  Let $X$ be a centered random variable with unit variance, zero third moment,
and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized
sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent
copies of $X$. Under these minimal conditions, we prove that $F_n$ converges in
distribution to a standard Gaussian random variable if and only if the
corresponding sequence of fourth moments converges to $3$. The statement is
then extended (mutatis mutandis) to the free probability setting. We shall also
discuss the optimality of our conditions in terms of explicit thresholds, as
well as establish several connections with the so-called universality
phenomenon of probability theory. Both in the classical and free probability
frameworks, our results extend and unify previous Fourth Moment Theorems for
Gaussian and semicircular approximations. Our techniques are based on a fine
combinatorial analysis of higher moments for homogeneous sums.
