  We consider the family of renormalizable scalar QFTs with self-interacting
potentials of highest monomial $\phi^{m}$ below their upper critical dimensions
$d_c=\frac{2m}{m-2}$, and study them using a combination of CFT constraints,
Schwinger-Dyson equation and the free theory behavior at the upper critical
dimension. For even integers $m \ge 4$ these theories coincide with the
Landau-Ginzburg description of multi-critical phenomena and interpolate with
the unitary minimal models in $d=2$, while for odd $m$ the theories are
non-unitary and start at $m=3$ with the Lee-Yang universality class. For all
the even potentials and for the Lee-Yang universality class, we show how the
assumption of conformal invariance is enough to compute the scaling dimensions
of the local operators $\phi^k$ and of some families of structure constants in
either the coupling's or the $\epsilon$-expansion. For all other odd potentials
we express some scaling dimensions and structure constants in the coupling's
expansion.
