  We study the base sizes of finite quasiprimitive permutation groups of
twisted wreath type, which are precisely the finite permutation groups with a
unique minimal normal subgroup that is also non-abelian, non-simple and
regular. Every permutation group of twisted wreath type is permutation
isomorphic to a twisted wreath product $G=T^k{:}P$ acting on its base group
$\Omega=T^k$, where $T$ is some non-abelian simple group and $P$ is some group
acting transitively on $\boldsymbol{k}=\{1,\ldots,k\}$ with $k\geq 2$. We prove
that if $G$ is primitive on $\Omega$ and $P$ is quasiprimitive on
$\boldsymbol{k}$, then $G$ has base size $2$. We also prove that the proportion
of pairs of points that are bases for $G$ tends to 1 as $|G|\to \infty$ when
$G$ is primitive on $\Omega$ and $P$ is primitive on $\boldsymbol{k}$. Lastly,
we determine the base size of any quasiprimitive group of twisted wreath type
up to four possible values (and three in the primitive case). In particular, we
demonstrate that there are many families of primitive groups of twisted wreath
type with arbitrarily large base sizes.
