Summary: Consider a frictionless surface $S$ in a gravitational field that need not be uniform. Given two points $A$ and $B$ on $S$, what curve is traced out by a particle that starts at $A$ and reaches $B$ in the shortest time? This paper considers this problem on simple surfaces such as surfaces of revolution and solves the problem two ways: First, we use conservation of mechanical energy and the Euler-Lagrange equation; second, we use geometrical optics and the eikonal equation. We conclude with a discussion of the relativistic effects at relativistic velocities.