  We generalize the classical coorbit space theory developed by Feichtinger and
Gr"ochenig to quasi-Banach spaces. As a main result we provide atomic
decompositions for coorbit spaces defined with respect to quasi-Banach spaces.
These atomic decompositions are used to prove fast convergence rates of best
$n$-term approximation schemes. We apply the abstract theory to time-frequency
analysis of modulation spaces $M^{p,q}_m$, $0< p,q \leq \infty$.
