The orbits of a Hamiltonian system on a fixed energy level can be viewed as geodesics of the corresponding Jacobi-Mauptertuis metric on the configuration space.  For systems of two degrees of freedom, this is a metric on the two-dimensional configuration space.  In this talk I will look at some simple examples from celestial mechanics, starting with the Kepler problem and moving on to the collinear and isosceles three-body problems.  I will look at the problem of visualizing the Kepler surface by embedding it in Euclidean space and discuss questions about length-minimizing geodesics for the three-body problems.