diff --git a/HISTORY.md b/HISTORY.md
index 777f3f32c..eca4be995 100644
--- a/HISTORY.md
+++ b/HISTORY.md
@@ -49,6 +49,8 @@ Removed the method `returned(::Model, values, keys)`; please use `returned(::Mod
 The method `DynamicPPL.init` (for implementing `AbstractInitStrategy`) now has a different signature: it must return a tuple of the generated value, plus a transform function that maps it back to unlinked space.
 This is a generalisation of the previous behaviour, where `init` would always return an unlinked value (in effect forcing the transform to be the identity function).
 
+Improved performance of transformations of univariate distributions' samples to and from their vectorised forms.
+
 ## 0.38.9
 
 Remove warning when using Enzyme as the AD backend.
diff --git a/Project.toml b/Project.toml
index 1b5e52492..74c5b1cc1 100644
--- a/Project.toml
+++ b/Project.toml
@@ -17,8 +17,10 @@ DifferentiationInterface = "a0c0ee7d-e4b9-4e03-894e-1c5f64a51d63"
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 DocStringExtensions = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
 InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240"
+InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 LogDensityProblems = "6fdf6af0-433a-55f7-b3ed-c6c6e0b8df7c"
+LogExpFunctions = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
 MacroTools = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
 OrderedCollections = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
 Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
@@ -62,10 +64,12 @@ DocStringExtensions = "0.9"
 EnzymeCore = "0.6 - 0.8"
 ForwardDiff = "0.10.12, 1"
 InteractiveUtils = "1"
+InverseFunctions = "0.1.17"
 JET = "0.9, 0.10, 0.11"
 KernelAbstractions = "0.9.33"
 LinearAlgebra = "1.6"
 LogDensityProblems = "2"
+LogExpFunctions = "0.3.29"
 MCMCChains = "6, 7"
 MacroTools = "0.5.6"
 MarginalLogDensities = "0.4.3"
diff --git a/src/utils.jl b/src/utils.jl
index 2d7b0404f..ddbc96742 100644
--- a/src/utils.jl
+++ b/src/utils.jl
@@ -1,3 +1,6 @@
+using LogExpFunctions: LogExpFunctions
+using InverseFunctions: InverseFunctions, inverse
+
 # singleton for indicating if no default arguments are present
 struct NoDefault end
 const NO_DEFAULT = NoDefault()
@@ -261,42 +264,6 @@ invlink_transform(dist) = inverse(link_transform(dist))
 # Helper functions for vectorize/reconstruct values #
 #####################################################
 
-"""
-    UnwrapSingletonTransform(input_size::InSize)
-
-A transformation that unwraps a singleton array, returning a scalar.
-
-The `input_size` field is the expected size of the input. In practice this only determines
-the number of indices, since all dimensions must be 1 for a singleton. `input_size` is used
-to check the validity of the input, but also to determine the correct inverse operation.
-
-By default `input_size` is `(1,)`, in which case `tovec` is the inverse.
-"""
-struct UnwrapSingletonTransform{InSize} <: Bijectors.Bijector
-    input_size::InSize
-end
-
-UnwrapSingletonTransform() = UnwrapSingletonTransform((1,))
-
-function (f::UnwrapSingletonTransform)(x)
-    if size(x) != f.input_size
-        throw(DimensionMismatch("Expected input of size $(f.input_size), got $(size(x))"))
-    end
-    return only(x)
-end
-
-function Bijectors.with_logabsdet_jacobian(f::UnwrapSingletonTransform, x)
-    return f(x), zero(LogProbType)
-end
-
-function Bijectors.with_logabsdet_jacobian(
-    inv_f::Bijectors.Inverse{<:UnwrapSingletonTransform}, x
-)
-    f = inv_f.orig
-    result = reshape([x], f.input_size)
-    return result, zero(LogProbType)
-end
-
 """
     ReshapeTransform(input_size::InSize, output_size::OutSize)
 
@@ -370,6 +337,178 @@ function Bijectors.with_logabsdet_jacobian(::Bijectors.Inverse{<:ToChol}, y)
     )
 end
 
+struct Only end
+struct NotOnly end
+(::Only)(x) = x[]
+(::NotOnly)(y) = [y]
+function Bijectors.with_logabsdet_jacobian(::Only, x::AbstractVector{T}) where {T<:Real}
+    return (x[], zero(T))
+end
+Bijectors.with_logabsdet_jacobian(::Only, x::AbstractVector) = (x[], zero(LogProbType))
+InverseFunctions.inverse(::Only) = NotOnly()
+InverseFunctions.inverse(::NotOnly) = Only()
+Bijectors.with_logabsdet_jacobian(::NotOnly, y::T) where {T<:Real} = ([y], zero(T))
+Bijectors.with_logabsdet_jacobian(::NotOnly, y) = ([y], zero(LogProbType))
+struct ExpOnly{L<:Real}
+    lower::L
+end
+(e::ExpOnly)(y::AbstractVector{<:Real}) = exp(y[]) + e.lower
+function Bijectors.with_logabsdet_jacobian(e::ExpOnly, y::AbstractVector{<:Real})
+    yi = y[]
+    x = exp(yi)
+    return (x + e.lower, yi)
+end
+InverseFunctions.inverse(e::ExpOnly) = LogVect(e.lower)
+struct LogVect{L<:Real}
+    lower::L
+end
+(l::LogVect)(x::Real) = [log(x - l.lower)]
+function Bijectors.with_logabsdet_jacobian(l::LogVect, x::Real)
+    logx = log(x - l.lower)
+    return ([logx], -logx)
+end
+InverseFunctions.inverse(l::LogVect) = ExpOnly(l.lower)
+struct TruncateOnly{L<:Real,U<:Real}
+    lower::L
+    upper::U
+end
+function (t::TruncateOnly)(y::AbstractVector{<:Real})
+    lbounded, ubounded = isfinite(t.lower), isfinite(t.upper)
+    return if lbounded && ubounded
+        ((t.upper - t.lower) * LogExpFunctions.logistic(y[])) + t.lower
+    elseif lbounded
+        exp(y[]) + t.lower
+    elseif ubounded
+        t.upper - exp(y[])
+    else
+        y[]
+    end
+end
+function Bijectors.with_logabsdet_jacobian(
+    t::TruncateOnly, y::AbstractVector{T}
+) where {T<:Real}
+    lbounded, ubounded = isfinite(t.lower), isfinite(t.upper)
+    return if lbounded && ubounded
+        bma = t.upper - t.lower
+        yi = y[]
+        res = (bma * LogExpFunctions.logistic(yi)) + t.lower
+        # TODO: Bijectors uses this:
+        #    absy = abs(yi)
+        #    return log(bma) - absy - (2 * log1pexp(-absy))
+        # Check if it's more numerically stable. Don't immediately see a reason why, but I
+        # assume there's a reason for it.
+        logjac = log(bma) + yi - (2 * LogExpFunctions.log1pexp(yi))
+        res, logjac
+    elseif lbounded
+        yi = y[]
+        exp(yi) + t.lower, yi
+    elseif ubounded
+        yi = y[]
+        t.upper - exp(yi), yi
+    else
+        y[], zero(T)
+    end
+end
+InverseFunctions.inverse(t::TruncateOnly) = UntruncateVect(t.lower, t.upper)
+
+struct UntruncateVect{L<:Real,U<:Real}
+    lower::L
+    upper::U
+end
+function (u::UntruncateVect)(x::Real)
+    lbounded, ubounded = isfinite(u.lower), isfinite(u.upper)
+    return [
+        if lbounded && ubounded
+            LogExpFunctions.logit((x - u.lower) / (u.upper - u.lower))
+        elseif lbounded
+            log(x - u.lower)
+        elseif ubounded
+            log(u.upper - x)
+        else
+            x
+        end,
+    ]
+end
+function Bijectors.with_logabsdet_jacobian(u::UntruncateVect, x::Real)
+    lbounded, ubounded = isfinite(u.lower), isfinite(u.upper)
+    return if lbounded && ubounded
+        bma = u.upper - u.lower
+        xma = x - u.lower
+        xma_over_bma = xma / bma
+        [LogExpFunctions.logit(xma_over_bma)], -log(xma_over_bma * (u.upper - x))
+    elseif lbounded
+        log_xma = log(x - u.lower)
+        [log_xma], -log_xma
+    elseif ubounded
+        log_bmx = log(u.upper - x)
+        [log_bmx], -log_bmx
+    else
+        return zero(x)
+    end
+end
+InverseFunctions.inverse(u::UntruncateVect) = TruncateOnly(u.lower, u.upper)
+
+for dist_type in [
+    Distributions.Cauchy,
+    Distributions.Chernoff,
+    Distributions.Gumbel,
+    Distributions.JohnsonSU,
+    Distributions.Laplace,
+    Distributions.Logistic,
+    Distributions.NoncentralT,
+    Distributions.Normal,
+    Distributions.NormalCanon,
+    Distributions.NormalInverseGaussian,
+    Distributions.PGeneralizedGaussian,
+    Distributions.SkewedExponentialPower,
+    Distributions.SkewNormal,
+    Distributions.TDist,
+]
+    @eval begin
+        from_linked_vec_transform(::$dist_type) = Only()
+        to_linked_vec_transform(::$dist_type) = NotOnly()
+    end
+end
+for dist_type in [
+    Distributions.BetaPrime,
+    Distributions.Chi,
+    Distributions.Chisq,
+    Distributions.Erlang,
+    Distributions.Exponential,
+    Distributions.FDist,
+    # Wikipedia's definition of the Frechet distribution allows for a location parameter,
+    # which could cause its minimum to be nonzero. However, Distributionsistributions.jl's `Frechet`
+    # does not implement this, so we can lump it in here.
+    Distributions.Frechet,
+    Distributions.Gamma,
+    Distributions.InverseGamma,
+    Distributions.InverseGaussian,
+    Distributions.Kolmogorov,
+    Distributions.Lindley,
+    Distributions.LogNormal,
+    Distributions.NoncentralChisq,
+    Distributions.NoncentralF,
+    Distributions.Rayleigh,
+    Distributions.Rician,
+    Distributions.StudentizedRange,
+    Distributions.Weibull,
+]
+    @eval begin
+        from_linked_vec_transform(d::$dist_type) = ExpOnly(minimum(d))
+        to_linked_vec_transform(d::$dist_type) = LogVect(minimum(d))
+    end
+end
+function to_linked_vec_transform(d::Distributions.ContinuousUnivariateDistribution)
+    return UntruncateVect(minimum(d), maximum(d))
+end
+function from_linked_vec_transform(d::Distributions.ContinuousUnivariateDistribution)
+    return TruncateOnly(minimum(d), maximum(d))
+end
+from_vec_transform(::Distributions.UnivariateDistribution) = Only()
+to_vec_transform(::Distributions.UnivariateDistribution) = NotOnly()
+from_linked_vec_transform(::DiscreteUnivariateDistribution) = Only()
+to_linked_vec_transform(::DiscreteUnivariateDistribution) = NotOnly()
+
 """
     from_vec_transform(x)
 
@@ -377,7 +516,7 @@ Return the transformation from the vector representation of `x` to original repr
 """
 from_vec_transform(x::AbstractArray) = from_vec_transform_for_size(size(x))
 from_vec_transform(C::Cholesky) = ToChol(C.uplo) ∘ ReshapeTransform(size(C.UL))
-from_vec_transform(::Real) = UnwrapSingletonTransform()
+from_vec_transform(::Real) = Only()
 
 """
     from_vec_transform_for_size(sz::Tuple)
@@ -395,7 +534,6 @@ Return the transformation from the vector representation of a realization from
 distribution `dist` to the original representation compatible with `dist`.
 """
 from_vec_transform(dist::Distribution) = from_vec_transform_for_size(size(dist))
-from_vec_transform(::UnivariateDistribution) = UnwrapSingletonTransform()
 from_vec_transform(dist::LKJCholesky) = ToChol(dist.uplo) ∘ ReshapeTransform(size(dist))
 
 struct ProductNamedTupleUnvecTransform{names,T<:NamedTuple{names}}
@@ -441,7 +579,7 @@ end
 # This function returns the length of the vector that the function from_vec_transform
 # expects. This helps us determine which segment of a concatenated vector belongs to which
 # variable.
-_input_length(from_vec_trfm::UnwrapSingletonTransform) = 1
+_input_length(::Only) = 1
 _input_length(from_vec_trfm::ReshapeTransform) = prod(from_vec_trfm.output_size)
 function _input_length(trfm::ProductNamedTupleUnvecTransform)
     return sum(_input_length ∘ from_vec_transform, values(trfm.dists))
@@ -477,19 +615,6 @@ function from_linked_vec_transform(dist::Distribution)
     f_vec = from_vec_transform(inverse(f_invlink), size(dist))
     return f_invlink ∘ f_vec
 end
-
-# UnivariateDistributions need to be handled as a special case, because size(dist) is (),
-# which makes the usual machinery think we are dealing with a 0-dim array, whereas in
-# actuality we are dealing with a scalar.
-# TODO(mhauru) Hopefully all this can go once the old Gibbs sampler is removed and
-# VarNamedVector takes over from Metadata.
-function from_linked_vec_transform(dist::UnivariateDistribution)
-    f_invlink = invlink_transform(dist)
-    f_vec = from_vec_transform(inverse(f_invlink), size(dist))
-    f_combined = f_invlink ∘ f_vec
-    sz = Bijectors.output_size(f_combined, size(dist))
-    return UnwrapSingletonTransform(sz) ∘ f_combined
-end
 function from_linked_vec_transform(dist::Distributions.ProductNamedTupleDistribution)
     return invlink_transform(dist)
 end