  We obtain a classification of simple modules with finite weight
multiplicities over basic classical map superalgebras. Any such module is
parabolic induced from a simple cuspidal bounded module over a cuspidal map
superalgebra. Further on, any simple cuspidal bounded module is isomorphic to
an evaluation module. As an application, we obtain a classification of all
simple Harish-Chandra modules for basic classical loop superalgebras. Extending
these results to affine Kac-Moody Lie superalgebras obtained by adding the
degree derivation, we construct a family of bounded simple modules of level
zero, and conjecture that all bounded simple cuspidal modules belong to this
family. Finally, we show that for affine Kac-Moody Lie superalgebras of type I
the Kac induction functor reduces the classification of all simple bounded
modules to the classification of the same class of modules over the even part,
whose classification is claimed by Dimitrov and Grantcharov.
