We will give a pedagogical introduction to the topological classification of insulators and superconductors with an emphasis on the correspondence between bulk topological invariants and protected gapless boundary modes. We will begin with a discussion of band topology in one dimension followed by the theory of the integer quantum Hall effect, time reversal invariant Z2 topological insulators in two and three dimensions and topological superconductors. We will discuss the general classification of gapped free fermion Hamiltonians, along with its generalization to include topological defects. We will conclude with a discussion of Majorana fermion states at the boundaries of topological superconductors and in superconductor - topological insulator heterostructure devices.