  Our universe contains a great number of extremely compact and massive objects
which are generally accepted to be black holes. Precise observations of orbital
motion near candidate black holes have the potential to determine if they have
the spacetime structure that general relativity demands. As a means of
formulating measurements to test the black hole nature of these objects,
Collins and Hughes introduced "bumpy black holes": objects that are almost, but
not quite, general relativity's black holes. The spacetimes of these objects
have multipoles that deviate slightly from the black hole solution, reducing to
black holes when the deviation is zero. In this paper, we extend this work in
two ways. First, we show how to introduce bumps which are smoother and lead to
better behaved orbits than those in the original presentation. Second, we show
how to make bumpy Kerr black holes -- objects which reduce to the Kerr solution
when the deviation goes to zero. This greatly extends the astrophysical
applicability of bumpy black holes. Using Hamilton-Jacobi techniques, we show
how a spacetime's bumps are imprinted on orbital frequencies, and thus can be
determined by measurements which coherently track a small orbiting body's
orbital phase. We find that weak-field orbits of bumpy black holes are modified
exactly as expected from a Newtonian analysis of a body with a prescribed
multipolar structure, reproducing well-known results from the celestial
mechanics literature. The impact of bumps on strong-field orbits is especially
strong, suggesting that this framework will allow observations to set robust
limits on the extent to which a spacetime's multipoles deviate from the black
hole expectation.
