  We investigate on the lattice the Yukawa models in 2 dimensions with Z(2) and
U(1) symmetries. These models reduce to the usual and chiral Gross-Neveu
models, respectively, when the kinetic and the selfcoupling terms of the scalar
field are turned off. The numerical data and mean field arguments suggest that,
at least for some range of the scalar field hopping parameter, fermion mass is
dynamically generated for arbitrarily weak Yukawa coupling. The models are
asymptotically free in this coupling, like the Gross-Neveu models, even when
the scalar quartic selfcoupling is strong.
