The theory of semigroups of tlinear operators is a very elegant piece of pure mathematics: it has a number of important applications, notably in the moderntheory of practical differental equations and in probability theory. The classicbooks in this area contain a great deal of material and may prove daunting to someone meeting the subject for the first time. This book provides a short and vey gentle introduction to the topic. No prior knowledge of functional analysis is assumed, and technicalities from Lebesgue integration and probability theory are kept to a minimum. Four examples are used throughout semigroups of translations, the Gauss-Weierstrass semigroup, the Cauchy-Poisson semigroup and semigroups of fractional integrals.