The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary differential equations (ODEs) are made using examples illustrating some unexpected and often surprising behaviours of the true and numerical solutions. The book briefly reviews the various approaches existing in the literature and develops an error and well-posedness analysis for general one-step and multistep methods. The continuous extensions of Runge-Kutta methods are presented in detail, which are useful for more general problems such as dense output and discontinuous equations. Some deeper insight into convergence and superconvergence is then carried out for DDEs with various kinds of delays. The stepsize control mechanism is developed on a firm mathematical basis. Classical results and an unconventional analysis of stability with respect to forcing term are reviewed for ODEs in view of the subsequent stability analysis for DDEs. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding investigations for the numerical methods are made. Reformulations of DDEs as partial differential equations and subsequent semi-discretization are described and compared with the classical approach. A list of available codes is provided.