This chapter concerns numerical methods for initial value problems, with emphasis given to first-order scalar equations, where the function f : ℝ2 → ℝ and the numbers t 0, γ ∈ ℝ are given. We say that a function y ℝ → ℝ is a solution of (5.1) on an interval I containing t 0 provided y'(t ) = f (t , y (t )) for all t ∈ I (y solves the differential equation) and y (t 0) = γ. We will primarily be concerned with approximating solutions on a finite interval [t 0,t F], and in this context we will use the notation throughout this chapter for functions g ∈ C [t 0,t F]. In Section 5.8, we consider first-order systems of equations, where the function f ∈ ℝ1+m → ℝm, the number t 0 ∈ ℝ, and vector γ ∈ ℝm are given, as well as higher-order scalar differential equations, such as where f ℝ3 → ℝ is given, which we will treat by recasting them as first-order systems. Section 5.1 might be skimmed or skipped by those who have recently had a course in differential equations.