  We give a full classification of the multi-charge supersymmetric $p$-brane
solutions in the massless and massive maximal supergravities in dimensions
$D\ge2$ obtained from the toroidal reduction of eleven-dimensional
supergravity. We derive simple universal rules for determining the fractions of
supersymmetry that they preserve. By reversing the steps of dimensional
reduction, the $p$-brane solutions become intersections of $p$-branes, NUTs and
waves in D=10 or D=11. Having classified the lower-dimensional $p$-branes, this
provides a classification of all the intersections in D=10 and D=11 where the
harmonic functions depend on the space transverse to all the individual
objects. We also discuss the structure of U-duality multiplets of $p$-brane
solutions, and show how these translate into multiplets of harmonic and
non-harmonic intersections.
