  We introduce a new concept of the so-called {\it composite wavelet
transforms}. These transforms are generated by two components, namely, a kernel
function and a wavelet function (or a measure). The composite wavelet
transforms and the relevant Calder\'{o}n-type reproducing formulas constitute a
unified approach to explicit inversion of the Riesz, Bessel, Flett, parabolic
and some other operators of the potential type generated by ordinary
(Euclidean) and generalized (Bessel) translations. This approach is exhibited
in the paper. Another concern is application of the composite wavelet
transforms to explicit inversion of the k-plane Radon transform on $\bbr^n$. We
also discuss in detail a series of open problems arising in wavelet analysis of
$L_p$-functions of matrix argument.
