Motivated by the recent discussions of the Penrose limit of AdS_5\times S^5,
we examine a more general class of supersymmetric pp-wave solutions of the type
IIB theory, with a larger number of non-vanishing structures in the self-dual
5-form. One of the pp-wave solutions can be obtained as a Penrose limit of a
D3/D3 intersection. In addition to 16 standard supersymmetries these
backgrounds always allow for supernumerary supersymmetries. The latter are in
one-to-one correspondence with the linearly-realised world-sheet
supersymmetries of the corresponding exactly-solvable type IIB string action.
The pp-waves provide new examples where supersymmetries will survive in a
T-duality transformation on the x^+ coordinate. The T-dual solutions can be
lifted to give supersymmetric deformed M2-branes in D=11. The deformed M2-brane
is dual to a three-dimensional field theory whose renormalisation group flow
runs from the conformal fixed point in the infra-red regime to a non-conformal
theory as the energy increases. At a certain intermediate energy scale there is
a phase transition associated with a naked singularity of the M2-brane. In the
ultra-violet limit the theory is related by T-duality to an exactly-solvable
massive IIB string theory.Comment: Latex, 23 pages. Typographical errors corrected, and references adde
