  We construct a family of infinite simple groups that we call \emph{twisted
Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups
$sV$ ($s\in\mathbb{N}$). We use twisted Brin-Thompson groups to prove a variety
of results regarding simple groups. For example, we prove that every finitely
generated group embeds quasi-isometrically as a subgroup of a two-generated
simple group, strengthening a result of Bridson. We also produce examples of
simple groups that contain every $sV$ and hence every right-angled Artin group,
including examples of type $\textrm{F}_\infty$ and a family of examples of type
$\textrm{F}_{n-1}$ but not of type $\textrm{F}_n$, for arbitrary
$n\in\mathbb{N}$. This provides the second known infinite family of simple
groups distinguished by their finiteness properties.
