  We outline basics of a new approach to transverse momentum dependence in hard
processes. As an illustration, we consider hard exclusive transition process
$\gamma^* \gamma \to \pi^0$ at the handbag level. Our starting point is
coordinate representation for matrix elements of operators (in the simplest
case, bilocal ${\cal O}(0,z)$) describing a hadron with momentum $p$. Treated
as functions of $(pz)$ and $z^2$, they are parametrized through virtuality
distribution amplitudes (VDA) $\Phi (x, \sigma)$, with $x$ being
Fourier-conjugate to $(pz)$ and $\sigma$ Laplace-conjugate to $z^2$. For
intervals with $z^+=0$, we introduce the transverse momentum distribution
amplitude (TMDA) $\Psi (x, k_\perp)$, and write it in terms of VDA $\Phi (x,
\sigma)$. The results of covariant calculations, written in terms of $\Phi (x,
\sigma)$ are converted into expressions involving $\Psi (x, k_\perp)$. Starting
with scalar toy models, we extend the analysis onto the case of spin-1/2 quarks
and QCD. We propose simple models for soft VDAs/TMDAs, and use them for
comparison of handbag results with experimental (BaBar and BELLE) data on the
pion transition form factor. We also discuss how one can generate
high-$k_\perp$ tails from primordial soft distributions.
