Geometric frustration a occurs in antiferromagnetic spin systems on lattices with a topology where the energy of each interacting pair of spins cannot be minimized individually. Examples are triangular, Kagome and Pyrochlor lattices. The latter have tetrahedral unit cells. A characteristic property of frustrated spin systems is the large degeneracy of classical ground states. Little is known exactly about these interesting physical systems. Even less is known about the corresponding quantum spin systems. The only knowledge about these complicated systems has so far been based on elaborate numerical simulations. With the help of sharp matrix inequalities and tricks of mathematical physics that stem from early work on quantum field theory, it is, however, possible to obtain interesting exact results about the ground state and excitations of frustrated quantum spin systems. The talk is based on joint work with Elliott Lieb. It includes a brief introduction to quantum spins.