The cohomology of arithmetic groups is a rich subject with links to topology and number theory. A classic example of such a group is the general linear group GL_N(Z) over the integers. A classic theorem of Borel computes the rational homology of this group in degree d does not depend on N for sufficiently large N. Moreover, Borel explicitly computed the homology in these cases. Work of Quillen, Charney, and van der Kallen implies that the integral cohomology is also independent of N. In this case, the actual groups are intimately related to the algebraic K-theory of Z. If one replaces GL_N(Z) by a congruence subgroup, then Borel's theorem still applies. However, the integral homology is no longer stable, even in degree one. This workshop concerns two recent approaches to get around this problem. The lecture series of Church and Farb will explain the notion of representation stability and FI-modules, while the lecture series of Calegari and Emerton will explain the notion of completed cohomology.