  We investigate multi-field multicritical scalar theories using CFT
constraints on two- and three-point functions combined with the Schwinger-Dyson
equation. This is done in general and without assuming any symmetry for the
models, which we just define to admit a Landau-Ginzburg description that
includes the most general critical interactions built from monomials of the
form $\phi_{i_1} \cdots \phi_{i_m}$. For all such models we analyze to the
leading order of the $\epsilon$-expansion the anomalous dimensions of the
fields and those of the composite quadratic operators. For models with even $m$
we extend the analysis to an infinite tower of composite operators of arbitrary
order. The results are supplemented by the computation of some families of
structure constants. We also find the equations which constrain the nontrivial
critical theories at leading order and show that they coincide with the ones
obtained with functional perturbative RG methods. This is done for the case
$m=3$ as well as for all the even models. We ultimately specialize to $S_q$
symmetric models, which are related to the $q$-state Potts universality class,
and focus on three realizations appearing below the upper critical dimensions
$6$, $4$ and $\frac{10}{3}$, which can thus be nontrivial CFTs in three
dimensions.
