Nerve theorems offer combinatorial characterisations of algebraic structures. Categorists have come up with nerve theorems for increasingly general classes of structures. The talk will consist of a gentle introduction to this theory, focusing on nerve theorems for so-called familial and analytic monads. The motivation for these monads is that they lend themselves well to defining structures graphically, in a suitable sense. Time permitting, we will touch upon work in progress towards defining higher-dimensional structures using this technology.