The Coleman integral is an analytic tool that serves as the p-adic analogue of the usual real-valued line integral. The theory of Coleman integration plays a prominent role in the arithmetic of curves and abelian varieties, allowing us to find torsion and rational points and giving us a means to compute p-adic heights and regulators. We describe algorithms for computing Coleman integrals on hyperelliptic curves as well as some applications of these algorithms. Throughout, we illustrate our methods with numerical examples, computed using our Sage implementation. This is joint work with Robert Bradshaw and Kiran Kedlaya.