  Due to considerable recent interest in the use of density matrices for a wide
variety of purposes, including quantum computation, we present a general method
for their parameterizations in terms of Euler angles. We assert that this is of
more fundamental importance than (as several people have remarked to us) ``just
another parameterization of the density matrix.'' There are several uses to
which this methodology can be put. One that has received particular attention
is in the construction of certain distinguished (Bures) measures on the $(n^2
-1)$-dimensional convex sets of $n \times n$ density matrices.
