diff --git a/library/alloc/src/task.rs b/library/alloc/src/task.rs
index 252e04a4105..5edc5796056 100644
--- a/library/alloc/src/task.rs
+++ b/library/alloc/src/task.rs
@@ -13,11 +13,9 @@ use crate::sync::Arc;
 ///
 /// This trait is a memory-safe and ergonomic alternative to constructing a
 /// [`RawWaker`]. It supports the common executor design in which the data used
-/// to wake up a task is stored in an [`Arc`][arc]. Some executors (especially
+/// to wake up a task is stored in an [`Arc`]. Some executors (especially
 /// those for embedded systems) cannot use this API, which is why [`RawWaker`]
 /// exists as an alternative for those systems.
-///
-/// [arc]: ../../std/sync/struct.Arc.html
 #[unstable(feature = "wake_trait", issue = "69912")]
 pub trait Wake {
     /// Wake this task.
diff --git a/library/alloc/src/vec.rs b/library/alloc/src/vec.rs
index 8ca0a0883cb..f8e009a9736 100644
--- a/library/alloc/src/vec.rs
+++ b/library/alloc/src/vec.rs
@@ -50,12 +50,9 @@
 //! v[1] = v[1] + 5;
 //! ```
 //!
-//! [`Vec<T>`]: ../../std/vec/struct.Vec.html
-//! [`new`]: ../../std/vec/struct.Vec.html#method.new
-//! [`push`]: ../../std/vec/struct.Vec.html#method.push
-//! [`Index`]: ../../std/ops/trait.Index.html
-//! [`IndexMut`]: ../../std/ops/trait.IndexMut.html
-//! [`vec!`]: ../../std/macro.vec.html
+//! [`Vec<T>`]: Vec
+//! [`new`]: Vec::new
+//! [`push`]: Vec::push
 
 #![stable(feature = "rust1", since = "1.0.0")]
 
@@ -278,22 +275,18 @@ use crate::raw_vec::RawVec;
 /// `Vec` does not currently guarantee the order in which elements are dropped.
 /// The order has changed in the past and may change again.
 ///
-/// [`vec!`]: ../../std/macro.vec.html
 /// [`get`]: ../../std/vec/struct.Vec.html#method.get
 /// [`get_mut`]: ../../std/vec/struct.Vec.html#method.get_mut
-/// [`Index`]: ../../std/ops/trait.Index.html
-/// [`String`]: ../../std/string/struct.String.html
-/// [`&str`]: ../../std/primitive.str.html
-/// [`Vec::with_capacity`]: ../../std/vec/struct.Vec.html#method.with_capacity
-/// [`Vec::new`]: ../../std/vec/struct.Vec.html#method.new
-/// [`shrink_to_fit`]: ../../std/vec/struct.Vec.html#method.shrink_to_fit
-/// [`capacity`]: ../../std/vec/struct.Vec.html#method.capacity
-/// [`mem::size_of::<T>`]: ../../std/mem/fn.size_of.html
-/// [`len`]: ../../std/vec/struct.Vec.html#method.len
-/// [`push`]: ../../std/vec/struct.Vec.html#method.push
-/// [`insert`]: ../../std/vec/struct.Vec.html#method.insert
-/// [`reserve`]: ../../std/vec/struct.Vec.html#method.reserve
-/// [owned slice]: ../../std/boxed/struct.Box.html
+/// [`String`]: crate::string::String
+/// [`&str`]: type@str
+/// [`shrink_to_fit`]: Vec::shrink_to_fit
+/// [`capacity`]: Vec::capacity
+/// [`mem::size_of::<T>`]: core::mem::size_of
+/// [`len`]: Vec::len
+/// [`push`]: Vec::push
+/// [`insert`]: Vec::insert
+/// [`reserve`]: Vec::reserve
+/// [owned slice]: Box
 #[stable(feature = "rust1", since = "1.0.0")]
 #[cfg_attr(not(test), rustc_diagnostic_item = "vec_type")]
 pub struct Vec<T> {
@@ -430,8 +423,8 @@ impl<T> Vec<T> {
     /// that nothing else uses the pointer after calling this
     /// function.
     ///
-    /// [`String`]: ../../std/string/struct.String.html
-    /// [`dealloc`]: ../../alloc/alloc/trait.GlobalAlloc.html#tymethod.dealloc
+    /// [`String`]: crate::string::String
+    /// [`dealloc`]: crate::alloc::GlobalAlloc::dealloc
     ///
     /// # Examples
     ///
@@ -658,7 +651,7 @@ impl<T> Vec<T> {
     ///
     /// Note that this will drop any excess capacity.
     ///
-    /// [owned slice]: ../../std/boxed/struct.Box.html
+    /// [owned slice]: Box
     ///
     /// # Examples
     ///
@@ -867,7 +860,7 @@ impl<T> Vec<T> {
     ///
     /// [`truncate`]: #method.truncate
     /// [`resize`]: #method.resize
-    /// [`extend`]: ../../std/iter/trait.Extend.html#tymethod.extend
+    /// [`extend`]: Extend::extend
     /// [`clear`]: #method.clear
     ///
     /// # Safety
@@ -1214,8 +1207,6 @@ impl<T> Vec<T> {
     /// Removes the last element from a vector and returns it, or [`None`] if it
     /// is empty.
     ///
-    /// [`None`]: ../../std/option/enum.Option.html#variant.None
-    ///
     /// # Examples
     ///
     /// ```
@@ -1480,7 +1471,6 @@ impl<T> Vec<T> {
     /// ```
     ///
     /// [`resize`]: #method.resize
-    /// [`Clone`]: ../../std/clone/trait.Clone.html
     #[stable(feature = "vec_resize_with", since = "1.33.0")]
     pub fn resize_with<F>(&mut self, new_len: usize, f: F)
     where
@@ -1590,8 +1580,6 @@ impl<T: Clone> Vec<T> {
     /// assert_eq!(vec, [1, 2]);
     /// ```
     ///
-    /// [`Clone`]: ../../std/clone/trait.Clone.html
-    /// [`Default`]: ../../std/default/trait.Default.html
     /// [`resize_with`]: #method.resize_with
     #[stable(feature = "vec_resize", since = "1.5.0")]
     pub fn resize(&mut self, new_len: usize, value: T) {
@@ -1655,9 +1643,7 @@ impl<T: Default> Vec<T> {
     /// ```
     ///
     /// [`resize`]: #method.resize
-    /// [`Default::default()`]: ../../std/default/trait.Default.html#tymethod.default
-    /// [`Default`]: ../../std/default/trait.Default.html
-    /// [`Clone`]: ../../std/clone/trait.Clone.html
+    /// [`Default::default()`]: Default::default
     #[unstable(feature = "vec_resize_default", issue = "41758")]
     #[rustc_deprecated(
         reason = "This is moving towards being removed in favor \
@@ -2338,7 +2324,6 @@ impl<T> Vec<T> {
     /// Note that `drain_filter` also lets you mutate every element in the filter closure,
     /// regardless of whether you choose to keep or remove it.
     ///
-    ///
     /// # Examples
     ///
     /// Splitting an array into evens and odds, reusing the original allocation: