Introduction; Contact Transformations In The Plane (Differential Equations * Point Transformations and Extended Point Transformations * Contact Transformations * The Directrix * The Contact Transformations of the Polarity * Contact Transformations and Differential Equations); Contact Transformations in Space (Extended Point Transformations * Definition of a Contact Transformation in Space * The Directrix Equations * Geometric Theory of Partial Differential Equations * Further Examples of Contact Transformations * The Apsidal Transformation); Special Contact Transformations (Special Contact Transformations * Determination of P(x,p) from X (x,p) * Characterization of the General Contact Transformation); Continuous Transformation Groups (One Parameter Groups of Transformations * One Parameter Families of Contact Transformations * Transformations of Canonical Differential Equations * The Theorems of Liouville and Poisson * The Theorems of Liouville and Poincare * The Euler-Lagrange Equations and a Generalized Variational Calculus * Partial Differential Equations and Canonical Systems * Integration of the Canonical System given a Complete Integral of the Partial Differential Equation * Multiparameter Families of Contact Transformations * The Infinitesimal Contact Transformations * Multiparameter Groups of Contact Transformations); Selected Applications (Introduction * Judicious Guessing and the Undamped Harmonic Oscillator * Action-Angle Variables * Separation of Variables * Perturbation Theory); Bibliography