  We introduce twisted Steinberg algebras over a commutative unital ring $R$.
These generalise Steinberg algebras and are a purely algebraic analogue of
Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff
groupoid $G$ and each locally constant $2$-cocycle $\sigma$ on $G$ taking
values in the units $R^\times$, we study the algebra $A_R(G,\sigma)$ consisting
of locally constant compactly supported $R$-valued functions on $G$, with
convolution and involution "twisted" by $\sigma$. We also introduce a
"discretised" analogue of a twist $\Sigma$ over a Hausdorff \'etale groupoid
$G$, and we show that there is a one-to-one correspondence between locally
constant $2$-cocycles on $G$ and discrete twists over $G$ admitting a
continuous global section. Given a discrete twist $\Sigma$ arising from a
locally constant $2$-cocycle $\sigma$ on an ample Hausdorff groupoid $G$, we
construct an associated twisted Steinberg algebra $A_R(G;\Sigma)$, and we show
that it coincides with $A_R(G,\sigma^{-1})$. Given any discrete field
$\mathbb{F}_d$, we prove a graded uniqueness theorem for
$A_{\mathbb{F}_d}(G,\sigma)$, and under the additional hypothesis that $G$ is
effective, we prove a Cuntz--Krieger uniqueness theorem and show that
simplicity of $A_{\mathbb{F}_d}(G,\sigma)$ is equivalent to minimality of $G$.
