  We prove several Tauberian theorems for regularizing transforms of
vector-valued distributions. The regularizing transform of $f$ is given by the
integral transform $M^{f}_{\varphi}(x,y)=(f\ast\varphi_{y})(x),$
$(x,y)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, with kernel
$\varphi_{y}(t)=y^{-n}\varphi(t/y)$. We apply our results to the analysis of
asymptotic stability for a class of Cauchy problems, Tauberian theorems for the
Laplace transform, the comparison of quasiasymptotics in distribution spaces,
and we give a necessary and sufficient condition for the existence of the trace
of a distribution on $\left\{x_0\right\}\times \mathbb R^m$. In addition, we
present a new proof of Littlewood's Tauberian theorem.
