In the recent years, there has been substantial activity connecting graph theory and group theory via the concept of a metric approximation of an infinite group by finite objects (groups or graphs), particularly around sofic groups. This lead to numerous results which describe approximation properties of the group (for instance, amenability or Haagerup property) in terms of geometric properties of its approximations (e.g. hyperfiniteness or coarse embeddability in a Hilbert space of a graph sequence). In this talk, I will describe these connections and some recent results around them.