The Fukaya category is an interesting invariant of a symplectic manifold. It is, at first sight, a rather fearsome thing: its objects are all Lagrangian submanifolds, an enormous and unruly set. Nevertheless, in certain circumstances one can find a finite set of Lagrangians which `generate' the category in an appropriate sense. I will explain a criterion, due to Abouzaid-Fukaya-Oh-Ohta-Ono, for when this happens. I will then explain a result, due to Tim Perutz and myself, which shows that this criterion is satisfied automatically in a large number of cases which arise naturally in the context of homological mirror symmetry. I hope it will inspire audience members with an interest in Heegaard Floer homology to try some computations in the Fukaya category of a symmetric product.