  These lectures give an introduction to the novel duality relating type IIB
string theory in a maximally supersymmetric plane-wave background to N=4, d=4,
U(N) Super Yang-Mills theory in a particular large N and large R-charge limit
due to Berenstein, Maldacena and Nastase. In the first part of these lectures
the duality is derived from the AdS/CFT correspondence by taking a Penrose
limit of the AdS_5 x S^5 geometry and studying the corresponding double-scaling
limit on the gauge theory side. The resulting free plane-wave superstring is
then quantized in light-cone gauge. On the gauge theory side of the
correspondence the composite Super Yang-Mills operators dual to string
excitations are identified, and it is shown how the string spectrum can be
mapped to the planar scaling dimensions of these operators. In the second part
of these lectures we study the correspondence at the interacting respectively
non-planar level. On the gauge theory side it is demonstrated that the large N
large R-charge limit in question preserves contributions from Feynman graphs of
all genera through the emergence of a new genus counting parameter - in
agreement with the string genus expansion for non-zero g_s. Effective quantum
mechanical tools to compute higher genus contributions to the scaling
dimensions of composite operators are developed and explicitly applied in a
genus one computation. We then turn to the interacting string theory side and
give an elementary introduction into light-cone superstring field theory in a
plane-wave background and point out how the genus one prediction from gauge
theory can be reproduced. Finally, we summarize the present status of the
plane-wave string/gauge theory duality.
