  This talk surveys recent work on the contribution of instantons to the
anomalous dimensions of BMN operators in $\calN=4$ supersymmetric Yang--Mills
theory and the corresponding non-perturbative contributions to the mass-matrix
of excited string states in maximally supersymmetric plane-wave string theory.
The dependence on the coupling constants and the impurity mode numbers in the
gauge theory and string theory are in striking agreement.
  [Presented by MBG at the Einstein Symposium, Bibliotecha Alexandrina, June
4--6 2005.]
