Symmetry analysis based on Lie group theory is the most important method for solving nlinear problems aside from numerical computation. The method can be used to find the symmetries of almost any system of differential equations and the kwledge of these symmetries can be used to reduce the complexity of physical problems governed by the equations. This is a broad, self-contained, introduction to the basics of symmetry analysis for first and second year graduate students in science, engineering and applied mathematics. Mathematica-based software for finding the Lie point symmetries and Lie-Backlund symmetries of differential equations is included on a CD along with more than forty sample tebooks illustrating applications ranging from simple, low order, ordinary differential equations to complex systems of partial differential equations. MathReader 4.0 is included to let the user read the sample tebooks and follow the procedure used to find symmetries.