This is the last of three volumes which present some of the most important tools of applied mathematics used in solving problems in mathematics, physics and computer science. It includes such areas as probability theory, operator calculus, representation theory, and special functions. This third volume - Representations of Lie Groups - answers some basic questions, such as: how can a Lie algebra given in matrix terms, or by prescribed commutation relations be realized so as to give an idea of what it `looks like'? A concrete theory is presented with emphasis on techniques suitable for efficient symbolic computing. Another question is: how do classical mathematical constructs interact with Lie structures? Here stochastic processes are taken as an example. The volume concludes with a section on output of the MAPLE program, which is available from Kluwer Academic Publishers on the Internet. This book is intended for pure and applied mathematicians and theoretical computer scientists. It is suitable for self study by researchers, as well as being appropriate as a text for a course or advanced seminar.