Mathematical biology is an interdisciplinary field that lies at the interface of mathematics and biology. Mathematics plays an important role at all levels of biological organization and regulation. My research is driven by a desire to understand the roles of stochasticity (noise), structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability and statistics, dynamical systems, and random graph theory, to shed light on biological issues while generating new mathematical questions. Stochasticity is fundamental to gene expression and ion channel gating in neurons; it also plays an underappreciated role in physiological regulation at larger scales, such as the regulation of sleep and wake behavior. In my research, I investigate the effects of stochasticity, as well as physiological structure (such as neuronal networks), and the constraints of evolution, on real-world biological dynamics. A particular challenge in this work is the need to bridge multiple spatial and temporal scales. Below I provide an overview of the field of mathematical biology and I describe my contributions to this field. I use methods such as stochastic models in genetics and stochastic processes on graphs to understand interlocking small and large facets of biological regulation.