The most common reason for approximating derivatives by finite differences is to apply these to solve ordinary and patrial differential equations – ODEs and PDEs, respectively. In the case of ODEs, many of the well-established (and seemingly quite different) procedures are immediately related to FD approximations – often more closely than may be apparent from how these methods are customarily described. Together with some basic convergence and stability theory, this chapter surveys a variety of ODE solvers, with emphasis on their FD connection and on the computational advantages that high-order accurate approximations can provide.