  Our assumption that spacetime is a continuum leads to many challenges in
mathematical physics. Singularities, divergent integrals and the like threaten
many of our favorite theories, from Newtonian gravity to classical
electrodynamics, quantum electrodynamics and the Standard Model. In general
relativity, singularities are intimately connected to some of the theory's most
dramatic successful predictions. We survey these problems and the large amount
of work that has gone into dealing with them.
