  It is shown that if T is stable unsuperstable, and aleph_1< lambda
=cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)<
mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g.
aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These
results are generalized to kappa =cf(kappa) < kappa (T) in the place of
aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa
< kappa (T) then there is a universal tree of height kappa +1 in cardinality
lambda .
