We study the limiting behavior of the discrete spectra of congruence subgroups of an irreducible arithmetic lattice in a semisimple Lie group $G$. Assuming that the subgroups in question do not contain any non-trivial elements of finite order, one expects their spectra to converge to the Plancherel measure of $G$ (the limit multiplicity property). We are able to prove this property for the lattices ${\rm SL} (n, \mathfrak{o}_F)$, where $F$ is a number field, and obtain conditional results in general. The focus lies on

the case of non-compact quotients, where the spectra have a continuous part.

There are two main parts of the proof, which is based on Arthur's trace formula. First, we prove some general results on congruence subgroups of arithmetic lattices and their intersections with conjugacy classes. Second, we reduce the control of the continuous spectrum to two conjectural properties of intertwining operators, one global and one local, which we can verify for the groups ${\rm GL} (n)$ and ${\rm SL} (n)$. This is joint work with Erez Lapid (Jerusalem) and Werner Müller (Bonn).