  We prove statistical limit laws for sequences of Birkhoff sums of the type
$\sum_{j=0}^{n-1}v_n\circ T_n^j$ where $T_n$ is a family of nonuniformly
hyperbolic transformations. The key ingredient is a new martingale-coboundary
decomposition for nonuniformly hyperbolic transformations which is useful
already in the case when the family $T_n$ is replaced by a fixed transformation
$T$, and which is particularly effective in the case when $T_n$ varies with
$n$. In addition to uniformly expanding/hyperbolic dynamical systems, our
results include cases where the family $T_n$ consists of intermittent maps,
unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally
forced dispersing billiards. As an application, we prove a homogenization
result for discrete fast-slow systems where the fast dynamics is generated by a
family of nonuniformly hyperbolic transformations.
