  We prove quenched versions of (i) a large deviations principle (LDP), (ii) a
central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for
non-autonomous dynamical systems. A key advance is the extension of the
spectral method, commonly used in limit laws for deterministic maps, to the
general random setting. We achieve this via multiplicative ergodic theory and
the development of a general framework to control the regularity of Lyapunov
exponents of \emph{twisted transfer operator cocycles} with respect to a twist
parameter. While some versions of the LDP and CLT have previously been proved
with other techniques, the local central limit theorem is, to our knowledge, a
completely new result, and one that demonstrates the strength of our method.
Applications include non-autonomous (piecewise) expanding maps, defined by
random compositions of the form $T_{\sigma^{n-1}\omega}\circ\cdots\circ
T_{\sigma\omega}\circ T_\omega$. An important aspect of our results is that we
only assume ergodicity and invertibility of the random driving
$\sigma:\Omega\to\Omega$; in particular no expansivity or mixing properties are
