  In this chapter we present transformation semigroups and their applications.
We begin with Klein's approach to geometry based on invariants of
transformation groups. Then we present symmetry groups in chemistry and in
classical mechanics. Next we introduce one-parameter semigroups of
transformations and their applications in ergodic theory. Our main subject are
one-parameter semigroups of operators, in particular stochastic semigroups. We
present general results on their existence and long-time behaviour. We also
give examples of one-parameter semigroups related to Markov chains, diffusion
and processes with jumps. We focus on the applications of semigroups of
operators in biology. Among other things, we study models of: DNA evolution;
growth of erythrocyte population; gene expression; cell cycle; the movement of
bacteria and insects. We also consider models with stochastic noise and
different population models.
