diff --git a/fuzz/Cargo.lock b/fuzz/Cargo.lock
index 06687e27..34f32ab1 100644
--- a/fuzz/Cargo.lock
+++ b/fuzz/Cargo.lock
@@ -139,7 +139,7 @@ checksum = "74765f6d916ee2faa39bc8e68e4f3ed8949b48cccdac59983d287a7cb71ce9c5"
 
 [[package]]
 name = "roaring"
-version = "0.11.0"
+version = "0.11.1"
 dependencies = [
  "bytemuck",
  "byteorder",
diff --git a/fuzz/fuzz_targets/against_croaring.rs b/fuzz/fuzz_targets/against_croaring.rs
index 2d6571af..5652c751 100644
--- a/fuzz/fuzz_targets/against_croaring.rs
+++ b/fuzz/fuzz_targets/against_croaring.rs
@@ -5,11 +5,12 @@ mod arbitrary_ops;
 use libfuzzer_sys::arbitrary::{self, Arbitrary};
 use libfuzzer_sys::fuzz_target;
 
-use crate::arbitrary_ops::{check_equal, Operation};
+use crate::arbitrary_ops::{check_equal, BitmapIteratorOperation, CRoaringIterRange, Operation};
 
 #[derive(Arbitrary, Debug)]
 struct FuzzInput<'a> {
     ops: Vec<Operation>,
+    iter_ops: Vec<BitmapIteratorOperation>,
     initial_input: &'a [u8],
 }
 
@@ -35,6 +36,14 @@ fuzz_target!(|input: FuzzInput| {
     }
     lhs_r.internal_validate().unwrap();
     rhs_r.internal_validate().unwrap();
+
+    let mut lhs_c_iter = CRoaringIterRange::new(&lhs_c);
+    let mut lhs_r_iter = lhs_r.iter();
+
+    for op in input.iter_ops {
+        op.apply(&mut lhs_c_iter, &mut lhs_r_iter);
+    }
+
     check_equal(&lhs_c, &lhs_r);
     check_equal(&rhs_c, &rhs_r);
 });
diff --git a/fuzz/fuzz_targets/arbitrary_ops/mod.rs b/fuzz/fuzz_targets/arbitrary_ops/mod.rs
index 03a00321..423b1b6e 100644
--- a/fuzz/fuzz_targets/arbitrary_ops/mod.rs
+++ b/fuzz/fuzz_targets/arbitrary_ops/mod.rs
@@ -95,6 +95,16 @@ pub enum BitmapBinaryOperation {
     AndNot,
 }
 
+#[derive(Arbitrary, Debug)]
+pub enum BitmapIteratorOperation {
+    Next,
+    NextBack,
+    AdvanceTo(Num),
+    AdvanceBackTo(Num),
+    Nth(Num),
+    NthBack(Num),
+}
+
 impl ReadBitmapOperation {
     pub fn apply(&self, x: &mut croaring::Bitmap, y: &mut roaring::RoaringBitmap) {
         match *self {
@@ -387,6 +397,104 @@ impl BitmapBinaryOperation {
     }
 }
 
+pub struct CRoaringIterRange<'a> {
+    cursor: croaring::bitmap::BitmapCursor<'a>,
+    empty: bool,
+    start: u32,
+    end_inclusive: u32,
+}
+
+impl<'a> CRoaringIterRange<'a> {
+    pub fn new(bitmap: &'a croaring::Bitmap) -> Self {
+        CRoaringIterRange {
+            cursor: bitmap.cursor(),
+            start: 0,
+            end_inclusive: u32::MAX,
+            empty: false,
+        }
+    }
+
+    fn next(&mut self) -> Option<u32> {
+        if self.empty {
+            return None;
+        }
+        self.cursor.reset_at_or_after(self.start);
+        let res = self.cursor.current().filter(|&n| n <= self.end_inclusive);
+        match res {
+            None => self.empty = true,
+            Some(n) if n == self.end_inclusive => self.empty = true,
+            Some(n) => self.start = n + 1,
+        }
+        res
+    }
+
+    fn next_back(&mut self) -> Option<u32> {
+        if self.empty {
+            return None;
+        }
+        self.cursor.reset_at_or_after(self.end_inclusive);
+        if self.cursor.current().is_none_or(|n| n > self.end_inclusive) {
+            self.cursor.move_prev();
+        }
+        let res = self.cursor.current().filter(|&n| n >= self.start);
+        match res {
+            None => self.empty = true,
+            Some(n) if n == self.start => self.empty = true,
+            Some(n) => self.end_inclusive = n - 1,
+        }
+        res
+    }
+
+    fn advance_to(&mut self, num: u32) {
+        self.start = self.start.max(num);
+    }
+
+    fn advance_back_to(&mut self, num: u32) {
+        self.end_inclusive = self.end_inclusive.min(num);
+    }
+
+    fn nth(&mut self, num: u32) -> Option<u32> {
+        for _ in 0..num {
+            _ = self.next();
+        }
+        self.next()
+    }
+
+    fn nth_back(&mut self, num: u32) -> Option<u32> {
+        for _ in 0..num {
+            _ = self.next_back();
+        }
+        self.next_back()
+    }
+}
+
+impl BitmapIteratorOperation {
+    pub fn apply(&self, x: &mut CRoaringIterRange, y: &mut roaring::bitmap::Iter) {
+        match *self {
+            BitmapIteratorOperation::Next => {
+                assert_eq!(x.next(), y.next());
+            }
+            BitmapIteratorOperation::NextBack => {
+                assert_eq!(x.next_back(), y.next_back());
+            }
+            BitmapIteratorOperation::AdvanceTo(n) => {
+                x.advance_to(n.0);
+                y.advance_to(n.0);
+            }
+            BitmapIteratorOperation::AdvanceBackTo(n) => {
+                x.advance_back_to(n.0);
+                y.advance_back_to(n.0);
+            }
+            BitmapIteratorOperation::Nth(n) => {
+                assert_eq!(x.nth(n.0), y.nth(n.0 as usize));
+            }
+            BitmapIteratorOperation::NthBack(n) => {
+                assert_eq!(x.nth_back(n.0), y.nth_back(n.0 as usize));
+            }
+        }
+    }
+}
+
 pub(crate) fn check_equal(c: &croaring::Bitmap, r: &roaring::RoaringBitmap) {
     let mut lhs = c.iter();
     let mut rhs = r.iter();
diff --git a/roaring/src/bitmap/store/bitmap_store.rs b/roaring/src/bitmap/store/bitmap_store.rs
index dbfdc61f..0503e655 100644
--- a/roaring/src/bitmap/store/bitmap_store.rs
+++ b/roaring/src/bitmap/store/bitmap_store.rs
@@ -515,6 +515,9 @@ impl<B: Borrow<[u64; BITMAP_LENGTH]>> BitmapIter<B> {
                 } else if cmp == Ordering::Equal {
                     self.value_back
                 } else {
+                    // New key is greater than original key and key_back, this iterator is now empty
+                    self.key = self.key_back;
+                    self.value = 0;
                     self.value_back = 0;
                     return;
                 }
diff --git a/roaring/src/bitmap/store/interval_store.rs b/roaring/src/bitmap/store/interval_store.rs
index f4ae015e..185a719e 100644
--- a/roaring/src/bitmap/store/interval_store.rs
+++ b/roaring/src/bitmap/store/interval_store.rs
@@ -657,13 +657,18 @@ impl<I: SliceIterator<Interval>> RunIter<I> {
             self.forward_offset = value;
         } else {
             self.intervals.next();
+            self.forward_offset = 0;
             return;
         }
-        if Some(self.forward_offset as u64)
-            >= self.intervals.as_slice().first().map(|f| f.run_len())
-        {
+        let only_interval = self.intervals.as_slice().len() == 1;
+        let total_offset = u64::from(self.forward_offset)
+            + if only_interval { u64::from(self.backward_offset) } else { 0 };
+        if Some(total_offset) >= self.intervals.as_slice().first().map(|f| f.run_len()) {
             self.intervals.next();
             self.forward_offset = 0;
+            if only_interval {
+                self.backward_offset = 0;
+            }
         }
     }
 
@@ -672,20 +677,26 @@ impl<I: SliceIterator<Interval>> RunIter<I> {
             self.backward_offset = value;
         } else {
             self.intervals.next_back();
+            self.backward_offset = 0;
             return;
         }
-        if Some(self.backward_offset as u64)
-            >= self.intervals.as_slice().last().map(|f| f.run_len())
-        {
+        let only_interval = self.intervals.as_slice().len() == 1;
+        let total_offset = u64::from(self.backward_offset)
+            + if only_interval { u64::from(self.forward_offset) } else { 0 };
+        if Some(total_offset) >= self.intervals.as_slice().last().map(|f| f.run_len()) {
             self.intervals.next_back();
             self.backward_offset = 0;
+            if only_interval {
+                self.forward_offset = 0;
+            }
         }
     }
 
     fn remaining_size(&self) -> usize {
-        (self.intervals.as_slice().iter().map(|f| f.run_len()).sum::<u64>()
-            - self.forward_offset as u64
-            - self.backward_offset as u64) as usize
+        let total_size = self.intervals.as_slice().iter().map(|f| f.run_len()).sum::<u64>();
+        let total_offset = u64::from(self.forward_offset) + u64::from(self.backward_offset);
+        debug_assert!(total_size >= total_offset);
+        total_size.saturating_sub(total_offset) as usize
     }
 
     /// Advance the iterator to the first value greater than or equal to `n`.
@@ -708,10 +719,25 @@ impl<I: SliceIterator<Interval>> RunIter<I> {
                 if let Some(value) = index.checked_sub(1) {
                     self.intervals.nth(value);
                 }
-                self.forward_offset = n - self.intervals.as_slice().first().unwrap().start;
+                let first_interval = self.intervals.as_slice().first().unwrap();
+                self.forward_offset = n - first_interval.start;
+                if self.intervals.as_slice().len() == 1
+                    && u64::from(self.forward_offset) + u64::from(self.backward_offset)
+                        >= first_interval.run_len()
+                {
+                    // If we are now the only interval, and we've now met the forward offset,
+                    // consume the final interval
+                    _ = self.intervals.next();
+                    self.forward_offset = 0;
+                    self.backward_offset = 0;
+                }
             }
             Err(index) => {
                 if index == self.intervals.as_slice().len() {
+                    // Consume the whole iterator
+                    self.intervals.nth(index);
+                    self.forward_offset = 0;
+                    self.backward_offset = 0;
                     return;
                 }
                 if let Some(value) = index.checked_sub(1) {
@@ -743,10 +769,25 @@ impl<I: SliceIterator<Interval>> RunIter<I> {
                 if let Some(value) = backward_index.checked_sub(1) {
                     self.intervals.nth_back(value);
                 }
-                self.backward_offset = self.intervals.as_slice().last().unwrap().end - n;
+                let last_interval = self.intervals.as_slice().last().unwrap();
+                self.backward_offset = last_interval.end - n;
+                if self.intervals.as_slice().len() == 1
+                    && u64::from(self.forward_offset) + u64::from(self.backward_offset)
+                        >= last_interval.run_len()
+                {
+                    // If we are now the only interval, and we've now met the forward offset,
+                    // consume the final interval
+                    _ = self.intervals.next_back();
+                    self.forward_offset = 0;
+                    self.backward_offset = 0;
+                }
             }
             Err(index) => {
                 if index == 0 {
+                    // Consume the whole iterator
+                    self.intervals.nth_back(self.intervals.as_slice().len());
+                    self.forward_offset = 0;
+                    self.backward_offset = 0;
                     return;
                 }
                 let backward_index = self.intervals.as_slice().len() - index;
@@ -778,12 +819,24 @@ impl<I: SliceIterator<Interval>> Iterator for RunIter<I> {
     }
 
     fn nth(&mut self, n: usize) -> Option<Self::Item> {
+        if n > usize::from(u16::MAX) {
+            // Consume the whole iterator
+            self.intervals.nth(self.intervals.as_slice().len());
+            self.forward_offset = 0;
+            self.backward_offset = 0;
+            return None;
+        }
         if let Some(skip) = n.checked_sub(1) {
             let mut to_skip = skip as u64;
             loop {
-                let to_remove = (self.intervals.as_slice().first()?.run_len()
-                    - self.forward_offset as u64)
-                    .min(to_skip);
+                let full_first_interval_len = self.intervals.as_slice().first()?.run_len();
+                let consumed_len = u64::from(self.forward_offset)
+                    + if self.intervals.as_slice().len() == 1 {
+                        u64::from(self.backward_offset)
+                    } else {
+                        0
+                    };
+                let to_remove = (full_first_interval_len - consumed_len).min(to_skip);
                 to_skip -= to_remove;
                 self.forward_offset += to_remove as u16;
                 self.move_next();
@@ -2401,7 +2454,7 @@ mod tests {
         iter.advance_to(800);
         assert_eq!(iter.next(), Some(800));
         iter.advance_to(u16::MAX);
-        assert_eq!(iter.next(), Some(801));
+        assert_eq!(iter.next(), None);
 
         let mut iter = interval_store.iter();
         iter.advance_to(100);
diff --git a/roaring/tests/iter.rs b/roaring/tests/iter.rs
index 05591681..ea3d5571 100644
--- a/roaring/tests/iter.rs
+++ b/roaring/tests/iter.rs
@@ -210,6 +210,31 @@ fn interleaved_bitmap() {
     assert!(outside_in(values).eq(outside_in(bitmap)));
 }
 
+#[test]
+fn run_nth_max() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..0x1_0000);
+    let mut iter = bitmap.iter();
+    assert_eq!(iter.nth(0x0_FFFF), Some(0x0_FFFF));
+    assert_eq!(iter.len(), 0);
+    #[allow(clippy::iter_nth_zero)]
+    {
+        assert_eq!(iter.nth(0), None);
+    }
+    assert_eq!(iter.next(), None);
+}
+
+#[test]
+fn run_nth_back_max() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..0x1_0000);
+    let mut iter = bitmap.iter();
+    assert_eq!(iter.nth_back(0x0_FFFF), Some(0));
+    assert_eq!(iter.len(), 0);
+    assert_eq!(iter.nth_back(0), None);
+    assert_eq!(iter.next_back(), None);
+}
+
 proptest! {
     #[test]
     fn interleaved_iter(values in btree_set(any::<u32>(), 50_000..=100_000)) {
diff --git a/roaring/tests/iter_advance_to.rs b/roaring/tests/iter_advance_to.rs
index d07ca671..8e3566b3 100644
--- a/roaring/tests/iter_advance_to.rs
+++ b/roaring/tests/iter_advance_to.rs
@@ -206,3 +206,95 @@ fn advance_bitset_back_to_start_word() {
     }
     assert_eq!(iter.next(), None);
 }
+
+#[test]
+fn advance_run_past_the_end() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..=0x35B00);
+    let mut iter = bitmap.iter();
+    iter.advance_to(0x35B01);
+    assert_eq!(iter.next(), None);
+}
+
+#[test]
+fn advance_run_back_before_start() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(500..=0x35B00);
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(499);
+    assert_eq!(iter.next_back(), None);
+}
+
+#[test]
+fn advance_run_back_reduces_forward_iter() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..=0x4000);
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(1);
+
+    assert_eq!(iter.next(), Some(0));
+    assert_eq!(iter.next(), Some(1));
+    assert_eq!(iter.next(), None);
+}
+
+#[test]
+fn advance_run_front_and_back_past_each_other() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..=0x4000);
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(100);
+    iter.advance_to(300);
+    assert_eq!(iter.next(), None);
+}
+
+#[test]
+fn advance_run_both_sides_past_each_other() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..0x1000);
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(100);
+    iter.advance_to(0xFFFF);
+    assert_eq!(iter.len(), 0);
+    assert_eq!(iter.nth_back(0), None);
+}
+
+#[test]
+fn advance_run_with_nth() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(36141..=224407);
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(101779);
+    assert_eq!(iter.nth(100563), None);
+}
+
+#[test]
+fn advance_to_with_next_len() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(100..0x4000);
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(100);
+    assert_eq!(iter.next(), Some(100));
+    assert_eq!(iter.len(), 0);
+    assert_eq!(iter.nth_back(0), None);
+}
+
+#[test]
+fn tmp() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(196363..=262143);
+    let mut iter = bitmap.iter();
+    assert_eq!(iter.next_back(), Some(262143));
+    iter.advance_to(228960);
+    assert_eq!(iter.nth(36643), None);
+}
+
+#[test]
+fn advance_bitset_front_and_back_past_each_other() {
+    let mut bitmap = RoaringBitmap::new();
+    bitmap.insert_range(0..=0x4000);
+    bitmap.remove_run_compression();
+    let mut iter = bitmap.iter();
+    iter.advance_back_to(100);
+    iter.advance_to(300);
+    assert_eq!(iter.next(), None);
+}