Ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) are widely used to model control systems in engineering, natural sciences, and economy. Optimal control plays a central role in optimizing such systems and to operate them effi ciently and safely. The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ODEs and DAEs. An emphasis is placed on the interplay between the optimal control problem, which typically is defi ned and analyzed in a Banach space setting, and discretizations thereof, which lead to finite dimensional optimization problems. The theoretical parts of the book require some knowledge of functional analysis, the numerically oriented parts require knowledge from linear algebra and numerical analysis. Practical examples are provided throughout the book for illustration purposes. The book addresses primarily master and PhD students as well as researchers in applied mathematics, but also engineers or scientists with a good background in mathematics. The book serves as a reference in research and teaching and hopefully helps to advance the state-of-the-art in optimal control.