diff --git a/docs/src/models/layers.md b/docs/src/models/layers.md
index 6230637744..e3d12a8f37 100644
--- a/docs/src/models/layers.md
+++ b/docs/src/models/layers.md
@@ -76,6 +76,8 @@ AlphaDropout
 LayerNorm
 InstanceNorm
 GroupNorm
+WeightNorm
+WeightNorm(::Dense)
 ```
 
 ### Testmode
diff --git a/src/Flux.jl b/src/Flux.jl
index b7d27406b0..df742b4a13 100644
--- a/src/Flux.jl
+++ b/src/Flux.jl
@@ -20,7 +20,7 @@ export Chain, Dense, Maxout, SkipConnection, Parallel, PairwiseFusion,
        RNN, LSTM, GRU, GRUv3,
        SamePad, Conv, CrossCor, ConvTranspose, DepthwiseConv,
        AdaptiveMaxPool, AdaptiveMeanPool, GlobalMaxPool, GlobalMeanPool, MaxPool, MeanPool,
-       Dropout, AlphaDropout, LayerNorm, BatchNorm, InstanceNorm, GroupNorm,
+       Dropout, AlphaDropout, LayerNorm, BatchNorm, InstanceNorm, GroupNorm, WeightNorm,
        Upsample, PixelShuffle,
        fmap, cpu, gpu, f32, f64,
        testmode!, trainmode!
@@ -49,6 +49,7 @@ include("layers/basic.jl")
 include("layers/conv.jl")
 include("layers/recurrent.jl")
 include("layers/normalise.jl")
+include("layers/transform.jl")
 include("layers/upsample.jl")
 include("layers/show.jl")
 
diff --git a/src/layers/transform.jl b/src/layers/transform.jl
new file mode 100644
index 0000000000..effe139814
--- /dev/null
+++ b/src/layers/transform.jl
@@ -0,0 +1,279 @@
+
+# "normalise.jl" contains Dropout etc, and Batchnorm etc. 
+# Since that file is quite long, WeightNorm is in a new file.
+
+"""
+    WeightNorm(layer, weight::Symbol; dims, init=false)
+
+Applies a reparameterisation to parameters in `layer`, such that `layer.weight` is normalised to
+```
+    layer.weight .* scale ./ norm(layer.weight; dims)
+```
+before being used by `layer`. The new trainable parameter `scale` has the same shape
+as `sum(layer.weight; dims)`, and is initialised to `1`.
+
+Can be stacked: `WeightNorm(WeightNorm(layer, :A; dims=1), :B, dims=1)` applies this
+to fields `layer.A` and `layer.B`. We could make `WeightNorm(layer, :A, :B; dims=1)` construct
+this, if it turns out to be useful.
+
+Keyword `init=true` is described below.
+
+# Example
+```
+julia> d1 = Dense(ones(2, 3), [0, 10], identity)
+Dense(3 => 2)       # 8 parameters
+
+julia> d1([1,2,3])
+2-element Vector{Float64}:
+  6.0
+ 16.0
+
+julia> wd1 = WeightNorm(d1)  # defaults for this layer type
+WeightNorm(Dense(3 => 2), :weight; dims=2)
+
+julia> wd1([1,2,3])
+2-element Vector{Float64}:
+  3.4641016151377553
+ 13.464101615137755
+
+julia> Flux.params(wd1)  # 3rd entry is new scale parameter
+Params([[1.0 1.0 1.0; 1.0 1.0 1.0], [0, 10], [1.0; 1.0;;]])
+
+julia> d2 = Dense(ones(2, 3) ./ sqrt.(sum(abs2, ones(2,3); dims=2)), [0, 10]);
+
+julia> d2([1,2,3])  # same result
+2-element Vector{Float64}:
+  3.4641016151377553
+ 13.464101615137755
+```
+
+# Lazy Initialisation
+
+The option `init=true` changes the scale and the layer's bias based on the first
+minibatch `x` seen. This makes more assumptions about the layer:
+* Layer must have a field called `:bias` or `:b`,
+  and a field called `:sigma` or `:σ` for the activation function.
+* Both the bias and the scale must be mutable arrays.
+* `dims` must be chosen such that `length(scale) == size(layer(x))[end-2]`,
+  interpreted as the channel dimension.
+
+# Example
+```
+julia> using Flux, Statistics
+
+julia> n1 = WeightNorm(Dense(3 => 2), init=true)  # no activation function
+WeightNorm(Dense(3 => 2), :weight; dims=2, init=true)
+
+julia> x1 = rand(Float32, 3, 10);  # batch of 10 input vectors
+
+julia> y1 = n1(x1);
+
+julia> n1  # has been changed
+WeightNorm(Dense(3 => 2), :weight; dims=2)
+
+julia> y1 == n1(x1)  # changes were applied before first output
+true
+
+julia> mean(y1, dims=2)  # batch mean is approximately zero
+2×1 Matrix{Float32}:
+ -9.536743f-8
+ -9.536743f-8
+
+julia> std(y1, dims=2)  # batch std is approximately 1
+2×1 Matrix{Float32}:
+ 1.0
+ 0.9999999
+
+julia> n2 = WeightNorm(Dense(3 => 2, relu), init=true);  # with activation function
+
+julia> mean(n2(x1), dims=2)  # the mean that is 0 is before the relu
+2×1 Matrix{Float32}:
+ 0.40149665
+ 0.35264373
+```
+
+# Reference
+
+Salimans & Kingma, _Weight Normalization_ (2016) https://arxiv.org/abs/1602.07868
+
+"""
+struct WeightNorm{field,dims,L,S<:AbstractArray} # <: LazyTransform
+  layer::L
+  scale::S
+  init::Base.RefValue{Bool}  # not implemented yet
+end
+
+using Flux
+using Flux: functor, @functor
+using ChainRulesCore: ignore_derivatives
+using Statistics: mean, std
+
+# Flux.@layer WeightNorm
+@functor WeightNorm
+_big_show(io::IO, w::WeightNorm, indent::Int64) = _layer_show(io, w, indent)
+
+# (w::WeightNorm)(x) = transform(w)(x)
+function (w::WeightNorm)(x)
+  ignore_derivatives() do
+    w.init[] && _weightnorm_init!(w, x)
+  end
+  transform(w)(x)
+end
+
+function transform(w::WeightNorm{which,dims}) where {which,dims}
+  fields, re = functor(transform(w.layer))
+  done = weightnorm(getfield(fields, which), w.scale, dims)
+  re(merge(fields, NamedTuple{(which,)}((done,))))
+end
+
+function WeightNorm(layer, which::Symbol; dims, init::Bool=false)
+  array = getfield(transform(layer), which)
+  array isa AbstractArray{<:Number} || error("WeightNorm must apply to a field which is an array of numbers")
+  iszero(array) && @warn "WeightNorm should not be applied to an all-zero array" which string(layer)
+  _scale = dims isa Colon ? sum(array; dims=1:ndims(array)) : sum(array; dims)
+  scale = map(one, _scale)
+  WeightNorm{which, dims, typeof(layer), typeof(scale)}(layer, scale, Ref(init))
+end
+
+function weightnorm(array::AbstractArray, scale::AbstractArray, dims)
+  n2 = sum(abs2, array; dims)
+  ε = eps(eltype(array))
+  @. array * scale / sqrt(n2 + ε)
+end
+
+"""
+    transform(w::WeightNorm)
+
+Takes a normalisation wrapper like `WeightNorm` and returns
+the original layer with newly adjusted weights. Recursive.
+
+(Could have a better name? Not necc. specific to WeightNorm.)
+"""
+transform(x) = x
+
+"""
+    _weightnorm_init!(w::WeightNorm, x)
+
+This is called on the first execution, when using `WeightNorm(...; init=true)`.
+It mutates `w.scale` and `transform(w).bias`, and sets flag `init=false`.
+"""
+function _weightnorm_init!(w::WeightNorm{which,dims}, x) where {which,dims}
+  # First, make a version of the layer without an activation function:
+  layer = transform(w)
+  fields, re = functor(layer)
+  nt = hasproperty(fields, :σ) ? (σ=identity,) : hasproperty(fields, :sigma) ? (sigma=identity,) : (;)
+  noact = re(merge(fields, nt))
+
+  # Second, apply that to the data. The relevant statistics are along all except the channel dims?
+  y = noact(x)
+  ndims(y) > 1 || begin @error "can't apply _weightnorm_init! to a single sample, need a batch. Doing nothing"; return end
+  rdims = ((1:ndims(y)-2)..., ndims(y))
+  vec_mean = vec(mean(y; dims=rdims))
+  vec_std = vec(std(y; dims=rdims))
+
+  # There is (by default) a scale per channel dimension. 
+  if length(w.scale) == length(vec_std)
+    vec(w.scale) .= 1 ./ vec_std
+  else
+    error("WeightNorm's lazy initialisation is confused about what dimensions `scale` corresponds to")
+  end
+
+  # There should always be one bias per channel dimension:
+  bias = hasproperty(fields, :bias) ? fields.bias : hasproperty(fields, :b) ? fields.b : error("WeightNorm(...; init=true) needs a bias field!")
+  bias isa AbstractVector || error("WeightNorm(...; init=true) needs a bias vector!")
+  if length(bias) == length(vec_std)
+    # This requires an array we can write into. We could change it to make WeightNorm a mutable struct instead.
+    bias .= .-vec_mean ./ vec_std
+  else
+    error("WeightNorm's lazy initialisation is confused about what dimensions bias lines up with")
+  end
+  
+  # Done! We mutated arrays within the underlying layer.
+  w.init[] = false
+  nothing
+end
+
+"""
+    WeightNorm(layer)
+
+For known layers, the defaults are as follows:
+* Apply normalisation to all weight-like fields, but never to bias
+* Choose `dims` so that `scale` has one entry per output channel.
+
+That is, `dims` is chosen to make `length(scale) == size(layer(x))[end-2]` for any
+input `x` for which `size(x)[end] == size(layer(x))[end]` is the number of batches.
+The same assumption is needed for deferred initialisation via `init=true`.
+
+# Examples
+
+```jldoctest
+julia> bi = Flux.Bilinear((3, 3) => 5);
+
+julia> WeightNorm(bi)
+WeightNorm(Bilinear(3 => 5), :weight; dims=(2, 3))
+
+julia> bi.weight |> size
+(5, 3, 3)
+
+julia> bi(randn(3, 32), randn(3, 32)) |> size  # 5 output channels
+(5, 32)
+
+julia> WeightNorm(bi).scale |> length
+5
+```
+
+Convolutional layers:
+
+```jldoctest
+julia> c4 = Conv((4, 4), 3 => 7, pad=1);  # conv layer for 2D colour images.
+
+julia> WeightNorm(c4)
+WeightNorm(Conv((4, 4), 3 => 7, pad=1), :weight; dims=(1, 2, 3))
+
+julia> c4.weight |> size
+(4, 4, 3, 7)
+
+julia> WeightNorm(c4).scale |> length  # 7 output channels
+7
+```
+"""
+WeightNorm(d::Dense; dims=2, kw...) = WeightNorm(d, :weight; dims, kw...)
+WeightNorm(d::Bilinear; dims=(2,3), kw...) = WeightNorm(d, :weight; dims, kw...)
+WeightNorm(d::Embedding; dims=2, kw...) = WeightNorm(d, :weight; dims, kw...)
+
+WeightNorm(s::Scale; dims=2, kw...) = WeightNorm(s, :scale; dims, kw...)
+
+_conv_wdims(c) = Tuple(1:ndims(c.weight)-1)
+WeightNorm(c::Conv; dims=_conv_wdims(c), kw...) = WeightNorm(c, :weight; dims, kw...)
+WeightNorm(c::CrossCor; dims=_conv_wdims(c), kw...) = WeightNorm(c, :weight; dims, kw...)
+WeightNorm(c::ConvTranspose; dims=_conv_wdims(c), kw...) = WeightNorm(c, :weight; dims, kw...)
+
+# WeightNorm(r::Flux.Recur, args...; kw...) = Flux.Recur(WeightNorm(r.cell, args...; kw...), r.state)
+# WeightNorm(r::Flux.RNNCell) =
+
+function WeightNorm(layer)
+  train = Optimisers._trainable(layer)::NamedTuple
+  list = filter(keys(train)) do s
+    s in (:bias, :b, :σ) && return false
+    train[s] isa AbstractArray{<:Number} || return false
+    return true
+  end
+  name = typeof(layer).name.name
+  isempty(list) && error("WeightNorm found no suitable fields to normalise in $name")
+  error("""WeightNorm does now know about laye $name.
+           You should probably define a method as like this,
+           with some field in $list,
+           and some default value for `dims`: 
+               Flux.WeightNorm(layer::$name; dims=default, init=false) = WeightNorm(layer, :field; dims, init)""")
+end
+
+function Base.show(io::IO, w::WeightNorm{which, dims}) where {which, dims}
+  print(io, "WeightNorm(")
+  Base.show(io, w.layer)
+  print(io, ", :", which, "; dims=", dims)
+  if w.init[]
+    print(io, ", ")
+    printstyled(io, "init=true", color=:magenta)
+  end
+  print(io, ")")
+end