  We present three-dimensional, {\it non-axisymmetric} distorted black hole
initial data which generalizes the axisymmetric, distorted, non-rotating
[Bernstein93a] and rotating [Brandt94a] single black hole data developed by
Bernstein, Brandt, and Seidel. These initial data should be useful for studying
the dynamics of fully 3D, distorted black holes, such as those created by the
spiraling coalescence of two black holes. We describe the mathematical
construction of several families of such data sets, and show how to construct
numerical solutions. We survey quantities associated with the numerically
constructed solutions, such as ADM masses, apparent horizons, measurements of
the horizon distortion, and the maximum possible radiation loss ($MRL$).
