This talk is a report on joint work with S. Molchanov. One set of results involves the behavior of sums of the form $\sum_{i=1}^{N(n)}\exp{\beta(\sum_{i=1}^n V_{ij}}$ where $V_{ij}$ are iid random variables. We identify rates of growth of $N(n)$ which give stable limit laws for properly normalized and centered sums, and other rates which give rise to a Central Limit Theorem holding for these sums. Another aspect of the work, which is related, considers sums of the form $\sum_{x \in Q_{L(n)}} u(n,x)$, where the function $u(t,x)$ is the solution of parabolic Anderson model and $Q_L$ is a box in $Z^d$ of radius L. Again the limit behavior of such sums depends on the rate of growth of $L(n)$. The results for this setting give a relation between intermittency and the so-called quenched-to-annealed transition.