Using a carefully chosen set of examples, we illustrate the importance and ubiquity of quantitative reasoning in the biological sciences. The examples range across many different levels of biological organization, from diseases through ecosystems, and the problems addressed range from basic to applied. In addition to the overall theme that mathematical and statistical approaches are essential for understanding biological systems, three particular and interacting mathematical themes emerge. First, nonlinearity is pervasive; second, inclusion of stochasticity is essential; and third, issues of scale are common to all applications of quantitative approaches. Future progress in understanding many biological systems will depend on continued applications and developments in these three areas, and on understanding how nonlinearity, stochasticity, and scale interact.