We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman integrals and p-adic heights on elliptic and hyperelliptic curves. Throughout, we give several numerical examples, and we conclude by showing how to use Coleman integrals to explicitly find integral points on hyperelliptic curves whose Jacobians have Mordell-Weil rank equal to their dimension.