Constructing non-arithmetic lattices

Every arithmetic subgroup of a semi-simple Lie group is a lattice. The converse is true except for the cases of SO(n,1) and SU(n,1). These groups are (finite covers of) the isometry groups of real and complex hyperbolic space. The problem is open for SU(n,1) with n at least 4.
For SU(2,1) and SU(3,1) non-arithmetic lattices are very rare. In this talk I will give the background to this problem, survey what is known and then describe a current project to construct non-arithmetic lattices in SU(2,1).