Mathematical biology offers a rich set of topics, challenges and interesting questions. It can motivate new mathematical problems and contribute to a better understanding of complicated biological systems, including those in cell biology. In this lecture, I will describe some of the research carried out in my group over the years, with contributions from many talented colleagues and trainees. I will focus mainly on relationships between local (PDE) and nonlocal variants of models that describe the internal structure and the interactions of cells. I will also survey work on reaction-diffusion systems associated with cell motility and recent experimental validation of some of our models.