diff --git a/Project.toml b/Project.toml
index 1274201c2..b36f894cb 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRules"
 uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
-version = "1.13.0"
+version = "1.14.0"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
@@ -11,10 +11,10 @@ RealDot = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
 
 [compat]
-ChainRulesCore = "1.10"
+ChainRulesCore = "1.11"
 ChainRulesTestUtils = "1"
 Compat = "3.35"
-FiniteDifferences = "0.12.8"
+FiniteDifferences = "0.12.20"
 JuliaInterpreter = "0.8"
 RealDot = "0.1"
 StaticArrays = "1.2"
diff --git a/src/rulesets/Base/array.jl b/src/rulesets/Base/array.jl
index 184a833f0..63ce70ae0 100644
--- a/src/rulesets/Base/array.jl
+++ b/src/rulesets/Base/array.jl
@@ -342,3 +342,165 @@ function rrule(::typeof(fill), x::Any, dims...)
     fill_pullback(Ȳ) = (NoTangent(), project(sum(Ȳ)), nots...)
     return fill(x, dims...), fill_pullback
 end
+
+#####
+##### `findmax`, `maximum`, etc.
+#####
+
+for findm in (:findmin, :findmax)
+    findm_pullback = Symbol(findm, :_pullback)
+
+    @eval function frule((_, xdot), ::typeof($findm), x; dims=:)
+        y, ind = $findm(x; dims=dims)
+        return (y, ind), Tangent{typeof((y, ind))}(xdot[ind], NoTangent())
+    end
+
+    @eval function rrule(::typeof($findm), x::AbstractArray; dims=:)
+        y, ind = $findm(x; dims=dims)
+        project = ProjectTo(x)
+        # This pullback is a lot like the one for getindex. Ideally they would probably be combined?
+        function $findm_pullback((dy, _))  # this accepts e.g. Tangent{Tuple{Float64, Int64}}(4.0, nothing)
+            dy isa AbstractZero && return (NoTangent(), NoTangent())
+            x_thunk = @thunk project(_zerolike_writeat(x, unthunk(dy), dims, ind))
+            x_ithunk = InplaceableThunk(x_thunk) do dx
+                if dims isa Colon
+                    view(dx, ind) .= view(dx, ind) .+ Ref(unthunk(dy))
+                else
+                    view(dx, ind) .= view(dx, ind) .+ unthunk(dy)  # this could be .+=, but not on Julia 1.0
+                end
+                dx
+            end
+            return (NoTangent(), x_ithunk)
+        end
+        return (y, ind), $findm_pullback
+    end
+end
+
+# This function is roughly `setindex!(zero(x), dy, inds...)`:
+
+function _zerolike_writeat(x::AbstractArray{<:Number}, dy, dims, inds...)
+    # It's unfortunate to close over `x`, but `similar(typeof(x), axes(x))` doesn't 
+    # allow `eltype(dy)`, nor does it work for many structured matrices.
+    dx = fill!(similar(x, eltype(dy), axes(x)), 0)
+    view(dx, inds...) .= dy  # possibly 0-dim view, allows dy::Number and dy::Array, and dx::CuArray
+    dx
+end
+function _zerolike_writeat(x::AbstractArray, dy, dims, inds...)
+    # Since we have `x`, we can also handle arrays of arrays.
+    dx = map(zero, x)
+    if dims isa Colon
+        view(dx, inds...) .= Ref(dy)
+    else
+        view(dx, inds...) .= dy
+    end
+    dx
+end
+
+# Allow for second derivatives, by writing rules for `_zerolike_writeat`;
+# these rules are the reason it takes a `dims` argument.
+
+function frule((_, _, dydot), ::typeof(_zerolike_writeat), x, dy, dims, inds...)
+    return _zerolike_writeat(x, dy, dims, inds...), _zerolike_writeat(x, dydot, dims, inds...)
+end
+
+function rrule(::typeof(_zerolike_writeat), x, dy, dims, inds...)
+    z = _zerolike_writeat(x, dy, dims, inds...)
+    function _zerolike_writeat_pullback(dz)
+        dx = sum(view(unthunk(dz), inds...); dims=dims)
+        nots = map(_ -> NoTangent(), inds)
+        return (NoTangent(), NoTangent(), dx, NoTangent(), nots...)
+    end
+    return z, _zerolike_writeat_pullback
+end
+
+# These rules for `maximum` pick the same subgradient as `findmax`:
+
+function frule((_, xdot), ::typeof(maximum), x; dims=:)
+    y, ind = findmax(x; dims=dims)
+    return y, xdot[ind]
+end
+
+function rrule(::typeof(maximum), x::AbstractArray; dims=:)
+    (y, _), back = rrule(findmax, x; dims=dims)
+    maximum_pullback(dy) = back((dy, nothing))
+    return y, maximum_pullback
+end
+
+function frule((_, xdot), ::typeof(minimum), x; dims=:)
+    y, ind = findmin(x; dims=dims)
+    return y, xdot[ind]
+end
+
+function rrule(::typeof(minimum), x::AbstractArray; dims=:)
+    (y, _), back = rrule(findmin, x; dims=dims)
+    minimum_pullback(dy) = back((dy, nothing))
+    return y, minimum_pullback
+end
+
+#####
+##### `extrema`
+#####
+
+function rrule(::typeof(extrema), x::AbstractArray{<:Number}; dims=:)
+    if dims isa Colon
+        return _extrema_colon(x)
+    else
+        return _extrema_dims(x, dims)
+    end
+end
+
+function _extrema_colon(x)
+    ylo, ilo = findmin(x)
+    yhi, ihi = findmax(x)
+    project = ProjectTo(x)
+    function extrema_pullback((dylo, dyhi))  # accepts Tangent
+        if (dylo, dyhi) isa Tuple{AbstractZero, AbstractZero}
+            return (NoTangent(), NoTangent())
+        end
+        # One argument may be AbstractZero here. Use promote_op because 
+        # promote_type allows for * as well as +, hence gives Any.
+        T = Base.promote_op(+, typeof(dylo), typeof(dyhi))
+        x_nothunk = let
+        # x_thunk = @thunk begin  # this doesn't infer
+            dx = fill!(similar(x, T, axes(x)), false)
+            view(dx, ilo) .= dylo
+            view(dx, ihi) .= view(dx, ihi) .+ dyhi
+            project(dx)
+        end
+        # x_ithunk = InplaceableThunk(x_thunk) do dx
+        #     view(dx, ilo) .= view(dx, ilo) .+ dylo
+        #     view(dx, ihi) .= view(dx, ihi) .+ dyhi
+        #     dx
+        # end
+        return (NoTangent(), x_nothunk)
+    end
+    return (ylo, yhi), extrema_pullback
+end
+
+function _extrema_dims(x, dims)
+    ylo, ilo = findmin(x; dims=dims)
+    yhi, ihi = findmax(x; dims=dims)
+    y = similar(ylo, Tuple{eltype(ylo), eltype(yhi)})
+    map!(tuple, y, ylo, yhi)  # this is a GPU-friendly version of collect(zip(ylo, yhi))
+    project = ProjectTo(x)
+    function extrema_pullback_dims(dy_raw)
+        dy = unthunk(dy_raw)
+        @assert dy isa AbstractArray{<:Tuple{Any,Any}}
+        # Can we actually get Array{Tuple{Float64,ZeroTangent}} here? Not sure.
+        T = Base.promote_op(+, eltype(dy).parameters...)
+        x_nothunk = let
+        # x_thunk = @thunk begin  # this doesn't infer
+            dx = fill!(similar(x, T, axes(x)), false)
+            view(dx, ilo) .= first.(dy)
+            view(dx, ihi) .= view(dx, ihi) .+ last.(dy)
+            project(dx)
+        end
+        # x_ithunk = InplaceableThunk(x_thunk) do dx
+        #     view(dx, ilo) .= first.(dy)
+        #     view(dx, ihi) .= view(dx, ihi) .+ last.(dy)
+        #     dx
+        # end
+        return (NoTangent(), x_nothunk)
+    end
+    return y, extrema_pullback_dims
+end
diff --git a/src/rulesets/Base/nondiff.jl b/src/rulesets/Base/nondiff.jl
index 0412ac451..dad5533f7 100644
--- a/src/rulesets/Base/nondiff.jl
+++ b/src/rulesets/Base/nondiff.jl
@@ -96,6 +96,8 @@
 @non_differentiable all(::Any, ::Any)
 @non_differentiable any(::Any)
 @non_differentiable any(::Any, ::Any)
+@non_differentiable argmax(::Any)
+@non_differentiable argmin(::Any)
 @non_differentiable ascii(::AbstractString)
 @non_differentiable axes(::Any)
 @non_differentiable axes(::Any, ::Any)
diff --git a/test/rulesets/Base/array.jl b/test/rulesets/Base/array.jl
index f601a0bb8..ff3e93931 100644
--- a/test/rulesets/Base/array.jl
+++ b/test/rulesets/Base/array.jl
@@ -198,3 +198,79 @@ end
     test_rrule(fill, 55 + 0.5im, 5)
     test_rrule(fill, 3.3, (3, 3, 3))
 end
+
+@testset "findmin & findmax" begin
+    # Forward
+    test_frule(findmin, rand(10))
+    test_frule(findmax, rand(10))
+    @test @inferred(frule((nothing, rand(3,4)), findmin, rand(3,4))) isa Tuple{Tuple{Float64, CartesianIndex}, Tangent}
+    @test @inferred(frule((nothing, rand(3,4)), findmin, rand(3,4), dims=1)) isa Tuple{Tuple{Matrix, Matrix}, Tangent}
+    @test_skip test_frule(findmin, rand(3,4)) # error from test_approx(actual::CartesianIndex{2}, expected::CartesianIndex{2}
+    @test_skip test_frule(findmin, rand(3,4), output_tangent = (rand(), NoTangent()))
+    @test_skip test_frule(findmin, rand(3,4), fkwargs=(dims=1,))
+    # These skipped tests might be fixed by https://github.com/JuliaDiff/FiniteDifferences.jl/issues/188
+
+    # Reverse
+    test_rrule(findmin, rand(10), output_tangent = (rand(), false))
+    test_rrule(findmax, rand(10), output_tangent = (rand(), false))
+    test_rrule(findmin, rand(5,3))
+    test_rrule(findmax, rand(5,3))
+    @test [0 0; 0 5] == @inferred unthunk(rrule(findmax, [1 2; 3 4])[2]((5.0, nothing))[2])
+    @test [0 0; 0 5] == @inferred unthunk(rrule(findmax, [1 2; 3 4])[2]((5.0, NoTangent()))[2])
+    
+    # Reverse with dims:
+    @test [0 0; 5 6] == @inferred unthunk(rrule(findmax, [1 2; 3 4], dims=1)[2](([5 6], nothing))[2])
+    @test [5 0; 6 0] == @inferred unthunk(rrule(findmin, [1 2; 3 4], dims=2)[2]((hcat([5,6]), nothing))[2])
+    test_rrule(findmin, rand(3,4), fkwargs=(dims=1,), output_tangent = (rand(1,4), NoTangent()))
+    test_rrule(findmin, rand(3,4), fkwargs=(dims=2,))
+
+    # Second derivatives
+    test_frule(ChainRules._zerolike_writeat, rand(2, 2), 5.0, :, CartesianIndex(2, 2))
+    test_rrule(ChainRules._zerolike_writeat, rand(2, 2), 5.0, :, CartesianIndex(2, 2))
+    @test_skip test_rrule(ChainRules._zerolike_writeat, rand(2, 2), 5.0, 1, [CartesianIndex(2, 1) CartesianIndex(2, 2)]  ⊢ NoTangent())  # MethodError: no method matching isapprox(::Matrix{Float64}, ::Float64; rtol=1.0e-9, atol=1.0e-9)
+    y, bk = rrule(ChainRules._zerolike_writeat, rand(2, 2), 5.0, 1, [CartesianIndex(2, 1) CartesianIndex(2, 2)])
+    @test y == [0 0; 5 5]
+    @test bk([1 2; 3 4]) == (NoTangent(), NoTangent(), [3 4], NoTangent(), NoTangent())
+end
+
+@testset "$imum" for imum in [maximum, minimum]
+    # Forward
+    test_frule(imum, rand(10))
+    test_frule(imum, rand(3,4))
+    test_frule(imum, rand(3,4), fkwargs=(dims=1,))
+    test_frule(imum, [rand(2) for _ in 1:3])
+    test_frule(imum, [rand(2) for _ in 1:3, _ in 1:4]; fkwargs=(dims=1,))
+
+    # Reverse
+    test_rrule(imum, rand(10))
+    test_rrule(imum, rand(3,4))
+    test_rrule(imum, rand(3,4), fkwargs=(dims=1,))
+    test_rrule(imum, rand(3,4,5), fkwargs=(dims=(1,3),))
+
+    # Arrays of arrays
+    test_rrule(imum, [rand(2) for _ in 1:3]; check_inferred=false)
+    test_rrule(imum, [rand(2) for _ in 1:3, _ in 1:4]; fkwargs=(dims=1,), check_inferred=false)
+
+    # Case which attains max twice -- can't use FiniteDifferences for this
+    res = imum == maximum ? [0,1,0,0,0,0] : [1,0,0,0,0,0]
+    @test res == @inferred unthunk(rrule(imum, [1,2,1,2,1,2])[2](1.0)[2])
+
+    # Structured matrix -- NB the minimum is a structral zero here
+    @test unthunk(rrule(imum, Diagonal(rand(3) .+ 1))[2](5.5)[2]) isa Diagonal
+    @test unthunk(rrule(imum, UpperTriangular(rand(3,3) .+ 1))[2](5.5)[2]) isa UpperTriangular{Float64}
+    @test_skip test_rrule(imum, Diagonal(rand(3) .+ 1)) # MethodError: no method matching zero(::Type{Any}), from fill!(A::SparseArrays.SparseMatrixCSC{Any, Int64}, x::Bool)
+end
+
+@testset "extrema" begin
+    test_rrule(extrema, rand(10), output_tangent = (rand(), rand()))
+    test_rrule(extrema, rand(3,4), fkwargs=(dims=1,), output_tangent = collect(zip(rand(1,4), rand(1,4))))
+    # Case where both extrema are the same index, to check accumulation:
+    test_rrule(extrema, rand(1), output_tangent = (rand(), rand()))
+    test_rrule(extrema, rand(1,1), fkwargs=(dims=2,), output_tangent = hcat((rand(), rand())))
+    test_rrule(extrema, rand(3,1), fkwargs=(dims=2,), output_tangent = collect(zip(rand(3,1), rand(3,1))))
+    # Double-check the forward pass
+    A = randn(3,4,5)
+    @test extrema(A, dims=(1,3)) == rrule(extrema, A, dims=(1,3))[1]
+    B = hcat(A[:,:,1], A[:,:,1])
+    @test extrema(B, dims=2) == rrule(extrema, B, dims=2)[1]
+end