  We consider several renormalizable, scale free models in three space-time
dimensions which involve scalar and spinor fields. The Yukawa couplings are
bilinear in both the spinor and scalar fields and the potential is of sixth
order in the scalar field. In a model with a single scalar field and a complex
Fermion field in three Euclidean dimensions, the couplings in the theory are
both asymptotically free. This property is not retained in 2+1 dimensional
Minkowski space, as we illustrate by considering a renormalizable scale-free
supersymmetric model. This is on account of the different properties of the
Dirac matrices in Euclidean and Minkowski space. We also examine a model in 2+1
dimensional Minkowski space in which two species of Fermions, associated with
the two unitarily inequivalent representations of the $2 \times 2$ Dirac
matrices, couple in two different ways to two distinct scalar fields. There are
two types of Yukawa couplings in this model, and either one or the other of
them can be asymptotically free (but not both simultaneously).
