It is a familiar fact (sometimes attributed to Ahlbrandt-Ziegler, though it is possibly older) that two aleph0-categorical theories are bi-interpretable if and only if their countable models have isomorphic topological isomorphism groups. Conversely, groups arising in this manner can be given an abstract characterisation, and a countable model of the theory (up to bi-interpretation, of course) can be reconstructed.
More recently, it was observed that various (interpretation-invariant) properties of an aleph0-categorical theory (stability, NIP, ...) correspond to familiar dynamical properties of the automorphism group, with a few nice applications.
Since stability and NIP have nothing to do with aleph0-categoricity, one may ask whether the correspondence between theories (up to bi-interpretation) and topological groups can be extended to theories which are not aleph0-categorical? There are some indications that a positive answer is possible, if one is willing to replace groups with more general objects.