The textbook provides a comprehensive and up-to-date presentation of the numerical treatment of initial value problems for ordinary differential equations and differential-algebraic systems. In addition to theoretical questions, practical aspects of the implementation and application of methods and software are discussed. The book is suitable for students of mathematics, computer science, physics, and engineering at universities and technical colleges. Knowledge of analysis, linear algebra, and numerical mathematics, as offered in foundational mathematics courses, is assumed. The content covers non-stiff differential equations - theoretical foundations, single-step methods, explicit extrapolation methods, linear multi-step methods, explicit peer methods, and the numerical comparison of non-stiff integrators as well as stiff differential equations.