  We consider the weighted Bergman spaces HL^2(B^d,\mu_{\lambda}), where
d\mu_\lambda(z)=c_{\lambda}(1-|z|^2)^lambda d\tau, \tau being the hyperbolic
volume measure. These spaces are nonzero if and only if \lambda>d. For
0<\lambda\leq d, spaces with the same formula for the reproducing kernel can be
defined using a Sobolev-type norm. We define Toeplitz operators on these
generalized Bergman spaces and investigate their properties. Specifically, we
describe classes of symbols for which the corresponding Toeplitz operators can
be defined as bounded operators or as a Hilbert--Schmidt operators on the
generalized Bergman spaces.
