Nonlinear semigroups are used within the field of applied mathematics and relate to problems as diverse as modelling traffic flows, neutron transport and coagulation. This text is an introductory treatment to the theory and applications of semigroups of operators, both linear and nonlinear. The first chapter introduces the requisite concepts from functional analysis. The second chapter presents the theory of semigroups of linear operators and applications to linear ACPs. The third extends these ideas to semilinear problems. The fourth and fifth chapters deal with fully nonlinear problems and indicate how some, but not all, of the results from the linear theory can be extended. The final chapter provides a variety of applications of practical interest. Exercises are included to help consolidate understanding.