This book addresses the question of the existence of oscillatory solutions to ODE's with delays. Oscillatory solutions are solutions with an infinite number of zeros in a given interval. In most cases the author treats a single equation. The delays can depend on time and may even become negative (advanced equations). The book is organized as follows. Chapter 1 summarizes the preliminaries; in fact, not much introduction is required, which makes the book accessible to non-specialists and students. Chapters 2 and 3 deal with first order equations. Chapter 4 reviews oscillation theory for second order equations without delays. These results are extended in chapters 5 and 6 to delayed equations of second order and higher orders respectively. While the previous chapters deal with single equations, chapter 7 is devoted to systems of two equations. The final chapter 8 introduces the reader to a new and interesting area of differential equations at time scales. Roughly speaking, here one studies an evolutionary equation with time varying over an arbitrary subset T (the “time scale”) of real numbers.