The asymptotic behaviour of elementary symmetric polynomials \(S_ n^{(k)}\) of order k, based on n independent and identically distributed random variables \(X_ 1,...,X_ n\), is investigated for the case that both k and n get large. If \(k=o(n^{1/2})\), then the distribution function of a suitably normalized \(S_ n^{(k)}\) is shown to converge to a standard normal limit. The speed of this convergence to normality is of order \(O(kn^{-})\), provided \[ k=O(\log ^{-1} n \log _ 2^{-1} n n^{1/2}) \] and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If \(k\to \infty\) at the rate \(n^{1/2}\) then a non-normal weak limit appears, provided the \(X_ j's\) are positive and \(S_ n^{(k)}\) is standardized appropriately. On the other hand, if \(k\to \infty\) at a rate faster than \(n^{1/2}\) then it is shown that for positive \(X_ j's\) there exists no linear norming which causes \(S_ n^{(k)}\) to converge weakly to a non degenerate weak limit.