In recent years ideas from additive combinatorics have made their way to group theory helping to settle a number of problems in group theory, geometry and number theory. For example, model theoretic ideas combined with a new look at the 1950's solution to Hilbert's fifth problem on locally compact groups, have lead to a structure theorem for approximate subgroups of arbitrary groups, yielding new proofs of Gromov's theorems on groups of polynomial growth and on almost flat manifolds. At the same time these ideas have been critical in establishing spectral gap bounds for Cayley graphs of finite groups with applications in analytic number theory and lattice point counting. The talk will be an introductory survey of some of these developments.