  In some respects the black hole plays the same role in gravitation that the
atom played in the nascent quantum mechanics. This analogy suggests that black
hole mass $M$ might have a discrete spectrum. I review the physical arguments
for the expectation that black hole horizon area eigenvalues are uniformly
spaced, or equivalently, that the spacing between stationary black hole mass
levels behaves like 1/M. This sort of spectrum has also emerged in a variety of
formal approaches to black hole quantization by a number of workers (with some
notable exceptions). If true, this result indicates a distortion of the
semiclassical Hawking spectrum which could be observable even for macroscopic
black holes. Black hole entropy suggests that the mentioned mass levels should
be degenerate to the tune of an exponential in $M^2$, as first noted by
Mukhanov. This has implications for the statistics of the radiation. I also
discuss open questions: whether radiative decay will spread the levels beyond
recognition, whether extremal black holes can be described by this scheme, etc.
I then describe an elementary algebra for the relevant black hole observables,
an outcome of work by Mukhanov and myself, which reproduces the uniformly
spaced area spectrum.
