diff --git a/NEWS.md b/NEWS.md
index 1ac663ea1be88..32c70cadc4d3d 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -364,6 +364,9 @@ Library improvements
     This supersedes the old behavior of reinterpret on Arrays. As a result, reinterpreting
     arrays with different alignment requirements (removed in 0.6) is once again allowed ([#23750]).
 
+  * `AbstractSet`s are now indexable, such that `set[x] == x` and `keys(set) == set`
+    ([#24508]).
+
 Compiler/Runtime improvements
 -----------------------------
 
diff --git a/base/bitset.jl b/base/bitset.jl
index 431df3f4ea9bf..2f8def248d121 100644
--- a/base/bitset.jl
+++ b/base/bitset.jl
@@ -15,7 +15,6 @@ very large integers), use [`Set`](@ref) instead.
 """
 BitSet(itr) = union!(BitSet(), itr)
 
-eltype(::Type{BitSet}) = Int
 similar(s::BitSet) = BitSet()
 copy(s1::BitSet) = copy!(BitSet(), s1)
 function copy!(dest::BitSet, src::BitSet)
@@ -23,7 +22,6 @@ function copy!(dest::BitSet, src::BitSet)
     copy!(dest.bits, src.bits)
     dest
 end
-eltype(s::BitSet) = Int
 sizehint!(s::BitSet, n::Integer) = (n > length(s.bits) && _resize0!(s.bits, n); s)
 
 # An internal function for setting the inclusion bit for a given integer n >= 0
diff --git a/base/set.jl b/base/set.jl
index a50f1fbc187dc..1b95851ebde40 100644
--- a/base/set.jl
+++ b/base/set.jl
@@ -1,5 +1,19 @@
 # This file is a part of Julia. License is MIT: https://julialang.org/license
 
+# AbstractSet
+
+# AbstractSets are idempotent under indexing:
+keys(set::AbstractSet) = set
+@inline function getindex(set::AbstractSet, x)
+    @boundscheck if x ∉ set
+        throw(KeyError(x))
+    end
+    return x
+end
+eltype(set::Type{<:AbstractSet{T}}) where {T} = T
+
+# Set
+
 mutable struct Set{T} <: AbstractSet{T}
     dict::Dict{T,Void}
 
@@ -23,9 +37,8 @@ function Set(g::Generator)
     return Set{T}(g)
 end
 
-eltype(::Type{Set{T}}) where {T} = T
-similar(s::Set{T}) where {T} = Set{T}()
-similar(s::Set, T::Type) = Set{T}()
+similar(s::AbstractSet) where {T} = similar(s, eltype(s))
+similar(s::AbstractSet, T::Type) = Set{T}() # default empty set type
 
 function show(io::IO, s::Set)
     print(io, "Set")
@@ -64,7 +77,7 @@ rehash!(s::Set) = (rehash!(s.dict); s)
 start(s::Set)       = start(s.dict)
 done(s::Set, state) = done(s.dict, state)
 # NOTE: manually optimized to take advantage of Dict representation
-next(s::Set, i)     = (s.dict.keys[i], skip_deleted(s.dict, i+1))
+@propagate_inbounds next(s::Set, i) = (s.dict.keys[i], skip_deleted(s.dict, i+1))
 
 """
     union(s1,s2...)
diff --git a/test/sets.jl b/test/sets.jl
index c237a6669759f..d20711b24c9ff 100644
--- a/test/sets.jl
+++ b/test/sets.jl
@@ -153,6 +153,12 @@ end
     end
 end
 
+@testset "indexing" begin
+    s = Set(["a", "b", "c"])
+    @test s["b"] == "b"
+    @test_throws KeyError s["d"]
+end
+
 @testset "union" begin
     @test isequal(union(Set([1])),Set([1]))
     s = ∪(Set([1,2]), Set([3,4]))