The Cauchy Problem for First Order Equations. Semigroup Theory. The Cauchy Problem for Second Order Equations. Cosine Function Theory. Reduction of a Second Order Equation to a First Order System. Phase Spaces. Applications to Partial Differential Equations. Uniformly Bounded Groups and Cosine Functions in Hilbert Space. The Parabolic Singular Perturbation Problems. Other Singular Perturbation Problems. The Complete Second Order Equation. Bibliography.