  We investigate possible renormalization-group fixed points at nonzero
coupling in $\phi^3$ theories in six spacetime dimensions, using beta functions
calculated to the four-loop level. We analyze three theories of this type, with
(a) a one-component scalar, (b) a scalar transforming as the fundamental
representation of a global ${\rm SU}(N)$ symmetry group, and (c) a scalar
transforming as a bi-adjoint representation of a global ${\rm SU}(N) \otimes
{\rm SU}(N)$ symmetry. We do not find robust evidence for such fixed points in
theories (a) or (b). Theory (c) has the special feature that the one-loop term
in the beta function is zero; implications of this are discussed.
