  In the field of quantum simulation of condensed matter phenomena by
artificially engineering the Hamiltonian of an atomic, molecular or optical
system, the concept of `synthetic dimensions' has recently emerged as a
powerful way to emulate phenomena such as topological phases of matter, which
are now of great interest across many areas of physics. The main idea of a
synthetic dimension is to couple together suitable degrees of freedom, such as
a set of internal atomic states, in order to mimic the motion of a particle
along an extra spatial dimension. This approach provides a way to engineer
lattice Hamiltonians and enables the realisation of higher-dimensional
topological models in platforms with lower dimensionality. We give an overview
of the recent progress in studying topological matter in synthetic dimensions.
After reviewing proposals and realizations in various setups, we discuss future
prospects in many-body physics, applications, and topological effects in three
or more spatial dimensions.
