Approximate groups are, roughly speaking, finite subsets of groups that are approximately closed under the group operations, such as the discrete interval {-N,...,N} in the integers.  Originally studied in arithmetic combinatorics, they also make an appearance in geometric group theory and in the theory of expansion in Cayley graphs. Hilbert's fifth problem asked for a topological description of Lie groups, and in particular whether any topological group that was a continuous (but not necessarily smooth) manifold was automatically a Lie group.  This problem was famously solved in the affirmative by Montgomery-Zippin and Gleason in the 1950s.
These two mathematical topics initially seem unrelated, but there is a remarkable correspondence principle (first implicitly used by Gromov, and later developed by Hrushovski and Breuillard, Green, and myself) that connects the combinatorics of approximate groups to problems in topological group theory such as Hilbert's fifth problem.  This correspondence has led to recent advances both in the understanding of approximate groups and in Hilbert's fifth problem, leading in particular to a classification theorem for approximate groups, which in turn has led to refinements of Gromov's theorem on groups of polynomial growth that have applications to the study of the topology of manifolds.  We will survey these interconnected topics in this talk.