We focus on the following problem, which we call “orbit recovery”: how many samples are required to estimate a signal when each sample has been acted on by a random element of a known, compact group? This question is motivated by and generalizes various “synchronization” problems, such as multi-reference alignment and the reconstruction problem from cryo-electon microscopy. Using tools from algebraic geometry and invariant theory, we give precise relationships between algebraic properties of the group action and the sample complexity of the statistical problem, under various success criteria. We also consider variations of this problem involving projection and heterogenous mixtures of signals. Based on joint work with Afonso S. Bandeira, Ben Blum-Smith, Amelia Perry, Philippe Rigollet, Amit Singer and Alexander S. Wein.