After introducing the concept of extra or synthetic dimension, i.e. encoding some of the spatial degrees of freedom as internal states of cold atoms, I will show how it offers the possibility of simulating:

i) 4d Hubbard models;
ii) artificial gauge fields in (semi-)synthetic 2d lattices;
iii) lattices on surfaces with non-trivial topology.

I will discuss which are the interesting observables characterizing the three families of quantum simulators and describe recent experimental progresses in their realization. I will conclude by presenting possible further applications of synthetic lattices.