Summary: Transversal homoclinic orbits of maps are known to generate shift dynamics on a set with Cantor-like structure. In this paper a numerical method is developed for computation of the corresponding homoclinic orbits. They are approximated by finite-orbit segments subject to asymptotic boundary conditions. We provide a detailed error analysis including a shadowing-type result by which one can infer the existence of a transversal homoclinic orbit from a finite segment. This approach is applied to several examples. In some of them parameters appear and closed loops of homoclinic orbits are found by a path-following algorithm.