  The chirality of electronic Bloch bands is responsible for many intriguing
properties of layered two-dimensional materials. We show that in bilayers of
transition metal dichalcogenides (TMDCs), unlike in few-layer graphene and
monolayer TMDCs, both intra-layer and inter-layer couplings give important
contributions to the Berry-curvature in the $K$ and $-K$ valleys of the
Brillouin zone. The inter-layer contribution leads to the stacking dependence
of the Berry curvature and we point out the differences between the commonly
available 3R type and 2H type bilayers. Due to the inter-layer contribution the
Berry curvature becomes highly tunable in double gated devices. We study the
dependence of the valley Hall and spin Hall effects on the stacking type and
external electric field. Although the valley and spin Hall conductivities are
not quantized, in MoS$_2$ 2H bilayers they may change sign as a function of the
external electric field which is reminiscent of the behaviour of lattice Chern
insulators.
