In this paper, we introduce recent developments in mathematical models for demography and epidemics. During last two decades, functional analytic approach for structured population dynamics has been rapidly developed, which was a result of fruitful interaction between applied mathematics and theoretical biology. First, we sketch some developments of idea of asynchronous exponential growth for age-structured populations, which is a modern refinement of traditional idea of Malthusian growth. Next general approach to non-linear structured population models is reviewed with special attention to pair formation models. Finally we consider a Kermack's and McKendrick's epidemic model and an age-structured SIR epidemic model to clear the crucial role of the basic reproduction number in the threshold phenomena. Though we know that many deterministic structured population models have been successfully examined as infinite dimensional dynamical system, even after efforts of two decades, building a universal framework to describe time evolution of structured populations under complex environments has been still an open problem.
