  We prove new velocity averaging results for second-order multidimensional
equations of the general form, $\op(\nabla_x,v)f(x,v)=g(x,v)$ where
$\op(\nabla_x,v):=\bba(v)\cdot\nabla_x-\nabla_x^\top\cdot\bbb(v)\nabla_x$.
These results quantify the Sobolev regularity of the averages,
$\int_vf(x,v)\phi(v)dv$, in terms of the non-degeneracy of the set $\{v:
|\op(\ixi,v)|\leq \delta\}$ and the mere integrability of the data, $(f,g)\in
(L^p_{x,v},L^q_{x,v})$. Velocity averaging is then used to study the
\emph{regularizing effect} in quasilinear second-order equations,
$\op(\nabla_x,\rho)\rho=S(\rho)$ using their underlying kinetic formulations,
$\op(\nabla_x,v)\chi_\rho=g_{{}_S}$. In particular, we improve previous
regularity statements for nonlinear conservation laws, and we derive completely
new regularity results for convection-diffusion and elliptic equations driven
by degenerate, non-isotropic diffusion.
