  Lattice studies of gauge theories with symplectic gauge groups provide
valuable information about gauge dynamics, and complement the results of
lattice investigations focused on unitary gauge groups. These theories play a
central role in phenomenological contexts such as composite Higgs and strongly
interacting dark matter models. We report on recent progress of our lattice
research programme, starting from the glueball mass spectrum and the topology
of the pure gauge theory. We present our results on the mass spectrum of mesons
in the quenched approximation, by varying the number of colours in the
symplectic group. For the $Sp(4)$ theory, we focus on results obtained with
dynamical fermion matter content comprising both fundamental and 2-index
antisymmetric representations of the gauge group, as dictated by a well known
model of composite Higgs with partial top compositeness.
