Atiyah-Singer index theorem relates analysis involving elliptic operators to geometry and topology of a closed manifold. A natural generalisation is to take into account of the symmetries of the space and study index theory of elliptic operators invariant under symmetries. The theory motivates significant developments in noncommutative geometry (e.g. Novikov conjecture). In this talk I will introduce index theory of invariant elliptic operators with several consequences in geometry and representation theory.