Since Charles Darwin’s seminal book on the origin of species, biologists and mathematicians have been studying the complex interaction between natural selection, mutation and recombination. The article explains these driving forces in evolution and illustrates with examples motivated from evolutionary biology how mathematical theory is developed from applied problems. We first explain with adaptive dynamics an example of a model which uses averaging principles for forces acting on separated time scales. Adaptive dynamics models the accumulation of beneficial mutations and the emergence of new species. We then consider RNA viruses which evolve so fast that the epidemiological and evolutionary time scales become the same. We describe different shapes of family trees (so called phylogenies) and explain that a good notion for capturing phylogenies is 0-hyperbolic metric spaces. We also discuss certain tree statistics that can be used to establish a notion of convergence of phylogenies. Finally, we explain with altruistic behaviour an example of cooperative selection and answer why altruistic behaviour may persist in natural populations.