  We study supersymmetric pp-waves in M-theory, their dimensional reduction to
D0-branes or pp-waves in type IIA, and their T-dualisation to solutions in the
type IIB theory. The general class of pp-waves that we consider encompass the
Penrose limits of AdS_p\times S^q with (p,q)=(4,7), (7,4), (3,3), (3,2), (2,3),
(2,2), but includes also many other examples that can again lead to
exactly-solvable massive strings, but which do not arise from Penrose limits.
All the pp-waves in D=11 have 16 ``standard'' Killing spinors, but in certain
cases one finds additional, or ``supernumerary,'' Killing spinors too. These
give rise to linearly-realised supersymmetries in the string or matrix models.
A focus of our investigation is on the circumstances when the Killing spinors
are independent of particular coordinates (x^+ or transverse-space
coordinates), since these will survive at the field-theory level in dimensional
reduction or T-dualisation.
