  : We consider two--dimensional supersymmetric Toda theories based on the Lie
superalgebras $A(n,n)$, $D(n+1,n)$ and $B(n,n)$ which admit a fermionic set of
simple roots and a fermionic untwisted affine extension. In particular, we
concentrate on two simple examples, the $B(1,1)$ and $A(1,1)$ theories. Both in
the conformal and massive case we address the issue of quantum integrability by
constructing the first non trivial conserved currents and proving their
conservation to all--loop orders. While the $D(n+1,n)$ and $B(n,n)$ systems are
genuine $N=1$ supersymmetric theories, the $A(n,n)$ models possess a global
$N=2$ supersymmetry. In the conformal case, we show that the $A(n,n)$
stress--energy tensor, uniquely determined by the holomorphicity condition, has
vanishing central charge and it corresponds to the stress--energy tensor of the
associated topological theory. (Invited talk at the International Workshop
``String theory, quantum gravity and the unification of the fundamental
interactions'', Roma, September 1992)
