  Two methods of constructing infinitely many isomorphically distinct $\Cal
L_p$-spaces have been published. In this article we show that these
constructions yield very different spaces and in the process develop methods
for dealing with these spaces from the isomorphic viewpoint. We use these
methods to give a complete isomorphic classification of the spaces $R_p^\alpha$
constructed by Bourgain, Rosenthal, and Schechtman and to show that $X_p\otimes
X_p$ is not isomorphic to a complemented subspace of any $R_p^\alpha.$
