A heteroclinic cycle is a structure in a dynamical system composed of a sequence of invariant sets---such as equilibria, periodic orbits, or even chaotic sets---and orbits which connect them in a cyclic manner. Near an attracting heteroclinic cycle, trajectories visit each invariant set in turn and, as time evolves, spend increasingly longer periods of time near each set, before making a rapid switch to the next one. A heteroclinic network is a connected union of heteroclinic cycles. Systems with heteroclinic cycles and networks have been invoked, for example, in models of intransitive competition between species and intermittent phenomena in fluid dynamics. They are also interesting from a dynamical systems perspective as a rich source of complicated behaviour, which we will discuss in this talk, and some work attempting to understand more about these structures.