Sequences of independent, identically distributed, (iid) random variables subject to mild moment conditions satisfy the central limit theorem (CLT) (convergence in distribution of the normalized averages to a normal distribution), the law of the iterated logarithm (LIL) (which quantifies, almost surely, the largest deviation of the normalized averages from the mean), and the almost sure invariance principle (ASIP). The ASIP is an almost sure approximation of partial sums by Brownian motion which implies the CLT and the LIL. Time-series of observations on a dynamical system may exhibit the same statistics. We will survey results on statistical properties of dynamical systems, such as the CLT and ASIP. We then describe recent work on establishing the ASIP for Hölder observations on a broad class of non-uniformly hyperbolic dynamical systems.