  Let $G \leqslant {\rm Sym}(\Omega)$ be a finite permutation group and recall
that the base size of $G$ is the minimal size of a subset of $\Omega$ with
trivial pointwise stabiliser. There is an extensive literature on base sizes
for primitive groups, but there are very few results for primitive groups of
product type. In this paper, we initiate a systematic study of bases in this
setting. Our first main result determines the base size of every product type
primitive group of the form $L \wr P \leqslant {\rm Sym}(\Omega)$ with soluble
point stabilisers, where $\Omega = \Gamma^k$, $L \leqslant {\rm Sym}(\Gamma)$
and $P \leqslant S_k$ is transitive. This extends recent work of Burness on
almost simple primitive groups. We also obtain an expression for the number of
regular suborbits of any product type group of the form $L \wr P$ and we
classify the groups with a unique regular suborbit under the assumption that
$P$ is primitive, which involves extending earlier results due to Seress and
Dolfi. We present applications on the Saxl graphs of base-two product type
groups and we conclude by establishing several new results on base sizes for
general product type primitive groups.
