A comprehensive introduction to the applications of symmetry analysis to differential equations. These applications, emerged from discoveries by Sophus Lie, can be used to find exact solutions and to verify and develop numerical schemes. The book includes both theoretical considerations and practical applications of use to physicists, chemists, mathematicians and engineers. There is a CD-ROM included with examples. The original installation instructions for the MathLie packages on the accompanying CD-ROM were found to have some problems for certain platforms. The electronic supplement by Eric Bynum and Gerd Baumann contains revised instructions that include more specific instructions for each of the major platforms and clear up some confusing sections of the original documentation.