The physiological properties of living cells are determined by underlying networks of interacting genes, mRNAs, proteins and metabolites. These biochemical networks are staggeringly complex, highly nonlinear, dynamical systems that process information in time and space, in order to determine the optimal responses of a cell to challenging environmental conditions and its own internal damage-reporting mechanisms. To ferret out these interactions is a problem for molecular cell biologists, but to understand how biochemical networks coordinate cellular responses is a problem in applied mathematics. Using some simple examples of cellular decision-making (bistable switches) and time-keeping (limit cycle oscillations), I will show how dynamical systems theory and numerical simulations can shed considerable light on the molecular basis of cell physiology.