  We provide explicit criteria for wavelets to give rise to frames and atomic
decompositions in ${\rm L}^2(\mathbb{R}^d)$, but also in more general Banach
function spaces. We consider wavelet systems that arise by translating and
dilating the mother wavelet, with the dilations taken from a suitable subgroup
of ${\rm GL}(\mathbb{R}^d)$, the so-called {\em dilation group}.The paper
provides a unified approach that is applicable to a wide range of dilation
groups, thus giving rise to new atomic decompositions for homogeneous Besov
spaces in arbitrary dimensions, but also for other function spaces such as
shearlet coorbit spaces.
  The atomic decomposition results are obtained by applying the coorbit theory
developed by Feichtinger and Gr\"ochenig, and they can be informally described
as follows: Given a function $\psi \in {\rm L}^2(\mathbb{R}^d)$ satisfying
fairly mild decay, smoothness and vanishing moment conditions, {\em any}
sufficiently fine sampling of the translations and dilations will give rise to
a wavelet frame. Furthermore, the containment of the analyzed signal in certain
smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by
looking at the frame coefficients, and convergence of the frame expansion holds
in the norms of these spaces. We motivate these results by discussing nonlinear
approximation.
