We discuss various still open questions on approximations of finitely generated groups, focusing on finite-dimensional approximations such as residual finiteness and soficity. We survey our results on the existence and stability of metric approximations. We suggest a few conjectures, e.g. on Gromov hyperbolic groups and their infinite monster limits. The setting is rather general and the involved concepts are transversal to various areas of mathematics. Based on joint works with Liviu Paunescu (Bucharest).