This talk will describe techniques for constructing the maximal subgroups of finite permutation groups. Apart from being interesting in themselves, the maximal subgroups of a group have many applications: for instance one may use them to investigate the full subgroup lattice, or to determine the character table of the group.

I will start by presenting the general approach taken by recent algorithms. We will then examine the techniques used to compute the maximal subgroups of various families of almost simple groups, before descibing recent work of Derek Holt and myself on constructing the maximal subgroups of finite black box classical groups.