Fix $\nu >0$, denote by $G(\nu/2)$ a Gamma random variable with parameter $\nu/2$, and let $n\geq2$ be an even integer. Consider a sequence $\{F_k\}_{k\geq 1}$ of square integrable random variables, belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\nu$. We prove that $\{F_k\}_{k\geq 1}$ converges in distribution to $2G(\nu/2) - \nu$, if, and only if, $E(F_k^4)-12 E(F_{k}^3)\rightarrow 12\nu^2-48\nu$. Observe that, if $\nu\geq 1$ is an integer, then $2G(\nu/2) - \nu$ has a centered $\chi^2$ law with $\nu$ degrees of freedom. Our approach involves the techniques of Malliavin calculus recently developed by Nualart and Ortiz-Latorre (2007). We also obtain some multidimensional non-central limit theorems, as well as several equivalent conditions in terms of Malliavin derivatives and norms of contraction operators. Our results should be compared with the main findings by Nualart and Peccati (2005), where it is shown that a normalized sequence of multiple Wiener-It\^{o} integrals converges in law to a Gaussian random variable if, and only if, the sequence of their fourth moments converges to 3.