Differential-algebraic equations (also called implicit equations or singular systems) are equations of the form F(t,x,x’)=0 that cannot be brought into an explicit form x’=f(t,x). Such equations form a classical topic going back to Kronecker and Weierstrass but this field still attracts a lot of research interest and it has many applications. Both the analytic and numerical study of an implicit equation revolves around the concept of the index. There are several ways of counting the index; the authors prefer the strangeness index, which measures ‘how far’ the given system is from a system of explicit ordinary differential equations decoupled with a purely algebraic system. The aim of the book is to present a self-contained and elementary introduction to the topic at the level of a graduate mathematics course. The first part of the book is devoted to analysis. Basic theory for linear problems with constant coefficients, problems with variable coefficients, and general nonlinear problems are developed. Though most of the exposition remains in the classical setting, generalized solutions and equations on manifolds are discussed as well. Some attention is given to control problems. The second part of the book is concerned with the numerical treatment of differential-algebraic equations. Runge-Kutta and BDF methods are developed first for systems with a low index while numerical index reduction is studied in later chapters. The final section gives an extensive list of available numerical packages.