Spatial dimensionality deeply affects what physical phenomena can emerge in a system. Although our physical world has only three spatial dimensions, new topological phases of matter are also predicted to emerge in four or more spatial dimensions. Now, this physics is becoming accessible thanks to advances in artificially-designed systems, such as photonics, cold atoms and electrical circuits. In this talk, I will review different methods for engineering “synthetic dimensions”, focusing particularly on how this can be used to realise synthetic photonic lattices. I will then discuss how recent progress in this direction is opening the way towards the simulation of topological matter in four spatial dimensions.