Is there a CLT-type result for $S^2$?

Question

The subject line is essentially the question: it there a theorem, like the CLT for the sample average, that allows us to say anything useful about the distribution of the sample variance as the sample size increases? I've seen simulations of the distribution of $S^2$ from distributions with high and low kurtosis as the sample sizes increase and they do appear to converge on the normal distribution. But I don't recall a theorem to that effect. I appreciate any help.