Humans are constant victims of infectious diseases. Biomedical research during this century has led to important insights into the molecular details of immune defense. Yet, many questions relating to disease require a quantitative understanding of the complex systems that arise from the nonlinear interactions between populations of immune cells and infectious agents. Exploration of such questions has lead to a newly emerging field of mathematical biology describing the spread of infectious agents both within and between infected individuals. This essay will discuss simple and complex models of evolution, and the propagation of virus and prion infections. Such models provide new perspectives for our understanding of infectious disease and provide guidelines for interpreting experimental observation; they also define what needs to be measured to improve understanding.