A finitely generated, residually finite group G is said to be profinitely rigid if the only finitely generated, residually finite groups with the same set of finite quotients as G are those that are isomorphic to G. More generally, one wants to know which properties P of groups are profinite invariants, i.e. if G has P and H has the same finite quotients as G, does H have P? For example, if G is a 3-manifold group, is H? If G is torsion-free, is H?

I will begin this talk with an overview of how the study of profinite rigidity has thrived in recent years due to a rich interplay between group theory, low-dimensional geometry and arithmetic. I'll then present recent work that underscores the importance of finiteness properties and the homology of groups in this context, describing results that exemplify extremes of rigid and non-rigid phenomena.