New infectious diseases emerge regularly, parasites mutate, and often become infectious again for the previously immunized host population. History provides many unfortunate examples. Abel and Riemann died from tuberculosis, Jacobi from smallpox, Kowalewskaja, Kummer and Weierstrass from influenza, and Boole and Descartes died from pneumonia. Fermat fortunately survived the last great outbreak of plague in Toulouse. Mathematics itself has successfully played a longstanding and important role in understanding the dynamics of epidemics. Due to the structural similarities of infection processes an abstract approach is natural. This has been realized and exploited a long time ago. In this talk some of the mathematical tools available for epidemiology are summarized and phenomena like epidemic waves and inter-epidemic periods, heterogeneous populations with variable infectivity, as well as vaccination schemes with subcritical bifurcations are discussed. Finally, we have a look at some actual data. Further information