  We review and present new results for a string moving on an $SU(1,1)$ group
manifold. We discuss two classes of theories which use discrete
representations. For these theories the representations forbidden by unitarity
decouple and, in addition, one can construct modular invariant partition
functions. The partion functions do, however, contain divergencies due to the
time-like direction of the $SU(1,1)$ manifold. The two classes of theories have
the corresponding central charges $c=9,6,5,9/2,\ldots$ and
$c=9,15,21,27,\ldots$. Subtracting two from the latter series of central
charges we get the Gervais-Neveu series $c-2=7,13,19,25$. This suggests a
relationship between the $SU(1,1)$ string and the Liouville theory, similar to
the one found in the $c=1$ string. Modular invariance is also demonstrated for
the principal continous representations. Furthermore, we present new results
for the Euclidean coset $SU(1,1)/U(1)$. The same two classes of theories will
be possible here and will have central charges $c=8,5,4,\dots$ and
$c=8,14,20,26,\ldots$, where the latter class includes the critical 2d black
hole. The partition functions for the coset theory are convergent.(Talk
presented by S.H. at the 16'th Johns Hopkins' Workshop, G\"oteborg, Sweden,
June 8-10, 1992)
