An approximate group is a group that is "almost closed" under multiplication. Finite approximate subgroups play a major role in additive combinatorics. Recently Breuillard, Green and Tao have established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov's polynomial growth theorem. Infinite approximate groups were studied implicitly long before the formal definition. Approximate subgroups of R^n that are Delone sets can be constructed using "cut-and-project" methods and are models for mathematical quasi-crystals. Recently, Björklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and their geometric aspects. This is joint work with Hartnick and Tonic.