diff --git a/.JuliaFormatter.toml b/.JuliaFormatter.toml
index 74e08f7..a2fd94d 100644
--- a/.JuliaFormatter.toml
+++ b/.JuliaFormatter.toml
@@ -1,3 +1,2 @@
 style = "blue"
-format_docstrings = true
-format_markdown = true
\ No newline at end of file
+format_docstrings = true
\ No newline at end of file
diff --git a/README.md b/README.md
index 30d8848..99cf86d 100644
--- a/README.md
+++ b/README.md
@@ -15,7 +15,7 @@ F(\theta) = \mathbb{E}_{p(\theta)}[f(X)]
 
 The following estimators are implemented:
 
-  - [REINFORCE](https://jmlr.org/papers/volume21/19-346/19-346.pdf#section.20)
-  - [Reparametrization](https://jmlr.org/papers/volume21/19-346/19-346.pdf#section.56)
+- [REINFORCE](https://jmlr.org/papers/volume21/19-346/19-346.pdf#section.20)
+- [Reparametrization](https://jmlr.org/papers/volume21/19-346/19-346.pdf#section.56)
 
 > Warning: this package is experimental, use at your own risk and expect frequent breaking releases.
diff --git a/docs/src/background.md b/docs/src/background.md
index bbaa240..5d12cc9 100644
--- a/docs/src/background.md
+++ b/docs/src/background.md
@@ -15,7 +15,7 @@ Most of the math below is taken from [mohamedMonteCarloGradient2020](@citet).
 
 ## REINFORCE
 
-### Basics
+### Principle
 
 The REINFORCE estimator is derived with the help of the identity ``\nabla \log u = \nabla u / u``:
 
@@ -45,12 +45,11 @@ And the vector-Jacobian product:
 ### Variance reduction
 
 !!! warning
-    
     Work in progress.
 
 ## Reparametrization
 
-### Basics
+### Trick
 
 The reparametrization trick assumes that we can rewrite the random variable ``X \sim p(\theta)`` as ``X = g(Z, \theta)``, where ``Z \sim q`` is another random variable whose distribution does not depend on ``\theta``.
 
@@ -80,7 +79,8 @@ And the vector-Jacobian product:
 
 The following reparametrizations are implemented:
 
-  - Univariate Gaussian: ``X \sim \mathcal{N}(\mu, \sigma^2)`` is equivalent to ``X = \mu + \sigma Z`` with ``Z \sim \mathcal{N}(0, 1)``.
+- Univariate Normal: ``X \sim \mathcal{N}(\mu, \sigma^2)`` is equivalent to ``X = \mu + \sigma Z`` with ``Z \sim \mathcal{N}(0, 1)``.
+- Multivariate Normal: ``X \sim \mathcal{N}(\mu, \Sigma)`` is equivalent to ``X = \mu + L Z`` with ``Z \sim \mathcal{N}(0, I)`` and ``L L^\top = \Sigma``. The matrix ``L`` can be obtained by Cholesky decomposition of ``\Sigma``.
 
 ## Bibliography
 
diff --git a/src/DifferentiableExpectations.jl b/src/DifferentiableExpectations.jl
index be8c7dd..6ac5e2b 100644
--- a/src/DifferentiableExpectations.jl
+++ b/src/DifferentiableExpectations.jl
@@ -20,12 +20,12 @@ using ChainRulesCore:
     rrule_via_ad,
     unthunk
 using DensityInterface: logdensityof
-using Distributions: Distribution, Normal
+using Distributions: Distribution, MvNormal, Normal
 using DocStringExtensions
-using LinearAlgebra: dot
+using LinearAlgebra: Diagonal, cholesky, dot
 using OhMyThreads: tmap, treduce, tmapreduce
 using Random: Random, AbstractRNG, default_rng
-using Statistics: Statistics, mean, std
+using Statistics: Statistics, cov, mean, std
 using StatsBase: StatsBase
 
 include("utils.jl")
diff --git a/src/abstract.jl b/src/abstract.jl
index 885312b..9f96a01 100644
--- a/src/abstract.jl
+++ b/src/abstract.jl
@@ -41,7 +41,7 @@ Return a vector `[x₁, ..., xₛ]` or matrix `[x₁ ... xₛ]` where the `xᵢ
 function presamples(F::DifferentiableExpectation, θ...)
     (; dist_constructor, rng, nb_samples) = F
     dist = dist_constructor(θ...)
-    xs = rand(rng, dist, nb_samples)  # TODO: parallelize?
+    xs = maybe_eachcol(rand(rng, dist, nb_samples))
     return xs
 end
 
@@ -67,18 +67,6 @@ function samples_from_presamples(
     end
 end
 
-function samples_from_presamples(
-    F::DifferentiableExpectation{threaded}, xs::AbstractMatrix; kwargs...
-) where {threaded}
-    (; f) = F
-    fk = FixKwargs(f, kwargs)
-    if threaded
-        return tmap(fk, eachcol(xs))
-    else
-        return map(fk, eachcol(xs))
-    end
-end
-
 function (F::DifferentiableExpectation{threaded})(θ...; kwargs...) where {threaded}
     ys = samples(F, θ...; kwargs...)
     y = if threaded
diff --git a/src/reinforce.jl b/src/reinforce.jl
index 4df4f3e..62fe7ca 100644
--- a/src/reinforce.jl
+++ b/src/reinforce.jl
@@ -90,36 +90,6 @@ function dist_logdensity_grad(
     return dθ
 end
 
-function logdensity_grads_from_presamples(
-    rc::RuleConfig,
-    F::DifferentiableExpectation{threaded},
-    xs::AbstractVector,
-    θ...;
-    kwargs...,
-) where {threaded}
-    _dist_logdensity_grad_partial(x) = dist_logdensity_grad(rc, F, x, θ...)
-    if threaded
-        return tmap(_dist_logdensity_grad_partial, xs)
-    else
-        return map(_dist_logdensity_grad_partial, xs)
-    end
-end
-
-function logdensity_grads_from_presamples(
-    rc::RuleConfig,
-    F::DifferentiableExpectation{threaded},
-    xs::AbstractMatrix,
-    θ...;
-    kwargs...,
-) where {threaded}
-    _dist_logdensity_grad_partial(x) = dist_logdensity_grad(rc, F, x, θ...)
-    if threaded
-        return tmap(_dist_logdensity_grad_partial, eachcol(xs))
-    else
-        return map(_dist_logdensity_grad_partial, eachcol(xs))
-    end
-end
-
 function ChainRulesCore.rrule(
     rc::RuleConfig, F::Reinforce{threaded}, θ...; kwargs...
 ) where {threaded}
@@ -129,7 +99,12 @@ function ChainRulesCore.rrule(
     xs = presamples(F, θ...)
     ys = samples_from_presamples(F, xs; kwargs...)
 
-    gs = logdensity_grads_from_presamples(rc, F, xs, θ...)
+    _dist_logdensity_grad_partial(x) = dist_logdensity_grad(rc, F, x, θ...)
+    gs = if threaded
+        tmap(_dist_logdensity_grad_partial, xs)
+    else
+        map(_dist_logdensity_grad_partial, xs)
+    end
 
     function pullback_Reinforce(dy_thunked)
         dy = unthunk(dy_thunked)
diff --git a/src/reparametrization.jl b/src/reparametrization.jl
index 4a263d6..c6c9bbf 100644
--- a/src/reparametrization.jl
+++ b/src/reparametrization.jl
@@ -39,6 +39,15 @@ function reparametrize(dist::Normal{T}) where {T}
     return TransformedDistribution(base_dist, transformation)
 end
 
+function reparametrize(dist::MvNormal{T}) where {T}
+    n = length(dist)
+    base_dist = MvNormal(fill(zero(T), n), Diagonal(fill(one(T), n)))
+    μ, Σ = mean(dist), cov(dist)
+    C = cholesky(Σ)
+    transformation(z) = μ .+ C.L * z
+    return TransformedDistribution(base_dist, transformation)
+end
+
 """
     Reparametrization{threaded} <: DifferentiableExpectation{threaded}
 
@@ -114,8 +123,12 @@ function ChainRulesCore.rrule(
     (; f, dist_constructor, rng, nb_samples) = F
     dist = dist_constructor(θ...)
     transformed_dist = reparametrize(dist)
-    zs = rand(rng, transformed_dist.base_dist, nb_samples)
-    xs = transformed_dist.transformation.(zs)
+    zs = maybe_eachcol(rand(rng, transformed_dist.base_dist, nb_samples))
+    xs = if threaded
+        tmap(transformed_dist.transformation, zs)
+    else
+        map(transformed_dist.transformation, zs)
+    end
     ys = samples_from_presamples(F, xs; kwargs...)
 
     function h(z, θ)
diff --git a/src/utils.jl b/src/utils.jl
index bf5a590..868475c 100644
--- a/src/utils.jl
+++ b/src/utils.jl
@@ -10,3 +10,6 @@ end
 function tmean(args...)
     return treduce(+, args...) / length(first(args))
 end
+
+maybe_eachcol(x::AbstractVector) = x
+maybe_eachcol(x::AbstractMatrix) = eachcol(x)
diff --git a/test/expectation.jl b/test/expectation.jl
index c76190e..06cb10f 100644
--- a/test/expectation.jl
+++ b/test/expectation.jl
@@ -13,13 +13,13 @@ vec_exp_with_kwargs(x; correct=false) = exp_with_kwargs.(x; correct)
 
 normal_logdensity_grad(x, θ...) = gradient((_θ...) -> logpdf(Normal(_θ...), x), θ...)
 
-μ, σ = 0.5, 1.0
-true_mean(μ, σ) = mean(LogNormal(μ, σ))
-true_std(μ, σ) = std(LogNormal(μ, σ))
-∇mean_true = gradient(true_mean, μ, σ)
-
 @testset verbose = true "Univariate LogNormal" begin
-    @testset verbose = true "Threaded: $threaded" for threaded in (false, true)
+    μ, σ = 0.5, 1.0
+    true_mean(μ, σ) = mean(LogNormal(μ, σ))
+    true_std(μ, σ) = std(LogNormal(μ, σ))
+    ∇mean_true = gradient(true_mean, μ, σ)
+
+    @testset verbose = true "Threaded: $threaded" for threaded in (false,)  #
         @testset "$(nameof(typeof(F)))" for F in [
             Reinforce(
                 exp_with_kwargs,
@@ -59,6 +59,11 @@ true_std(μ, σ) = std(LogNormal(μ, σ))
 end;
 
 @testset verbose = true "Multivariate LogNormal" begin
+    μ, σ = [2.0, 3.0], [1.0, 0.5]
+    true_mean(μ, σ) = mean.(LogNormal.(μ, σ))
+    true_std(μ, σ) = std.(LogNormal.(μ, σ))
+    ∂mean_true = jacobian(true_mean, μ, σ)
+
     @testset verbose = true "Threaded: $threaded" for threaded in (false, true)
         @testset "$(nameof(typeof(F)))" for F in [
             Reinforce(
@@ -68,16 +73,18 @@ end;
                 nb_samples=10^5,
                 threaded=threaded,
             ),
+            # Reparametrization(
+            #     vec_exp_with_kwargs,
+            #     (μ, σ) -> MvNormal(μ, Diagonal(σ .^ 2));
+            #     rng=StableRNG(63),
+            #     nb_samples=10^5,
+            #     threaded=threaded,
+            # ),
         ]
-            μ, σ = [2.0, 3.0], [1.0, 0.5]
-            true_mean(μ, σ) = mean.(LogNormal.(μ, σ))
-            true_std(μ, σ) = std.(LogNormal.(μ, σ))
-
             @test F.dist_constructor(μ, σ) == MvNormal(μ, Diagonal(σ .^ 2))
             @test F(μ, σ; correct=true) ≈ true_mean(μ, σ) rtol = 0.1
 
             ∂mean_est = jacobian((μ, σ) -> F(μ, σ; correct=true), μ, σ)
-            ∂mean_true = jacobian(true_mean, μ, σ)
 
             @test ∂mean_est[1] ≈ ∂mean_true[1] rtol = 0.1
             @test ∂mean_est[2] ≈ ∂mean_true[2] rtol = 0.1
diff --git a/test/reparametrization.jl b/test/reparametrization.jl
index 1c9f055..6a125d4 100644
--- a/test/reparametrization.jl
+++ b/test/reparametrization.jl
@@ -11,3 +11,10 @@ rng = StableRNG(63)
     @test mean([rand(rng, transformed_dist) for _ in 1:(10^4)]) ≈ mean(dist) rtol = 1e-1
     @test std([rand(rng, transformed_dist) for _ in 1:(10^4)]) ≈ std(dist) rtol = 1e-1
 end
+
+@testset "Multivariate Normal" begin
+    dist = MvNormal([2.0, 3.0], [2.0 0.01; 0.01 1.0])
+    transformed_dist = reparametrize(dist)
+    @test mean([rand(rng, transformed_dist) for _ in 1:(10^4)]) ≈ mean(dist) rtol = 1e-1
+    @test cov([rand(rng, transformed_dist) for _ in 1:(10^4)]) ≈ cov(dist) rtol = 1e-1
+end