In this talk I will give an introduction to the mathematical theory of strongly disordered topological insulators. These are novel materials which insulate in their bulk but (may) conduct along their edge; the quintessential example is that of the integer quantum Hall effect. What characterizes these materials is the existence of a topological index, experimentally measurable and macroscopically quantized. Mathematically this is explained by applying algebraic topology to the space of appropriate quantum mechanical Hamiltonians. I will survey some recent results regarding bulk-edge correspondence, calculation of the path-connected components and the connection to Anderson localization.