  We present a family of explicit formulae for evaluating absolute moments of
probability measures on $\mathbb{R}^d$ in terms of Fourier transforms. As to
the space of probability measures possessing finite absolute moments of an
arbitrary order, we exploit our formulae to characterize its Fourier image and
construct Fourier-based probability metrics which make the space complete. As
applications, we compute absolute moments of those probability measures whose
characteristic functions belong to the Scheonberg classes, estimate absolute
moments of convolutions and investigate the asymptotic behavior of solutions to
the heat-diffusion equations from a probability view-point.
