  This paper introduces methods for classifying actions of finite-dimensional
Hopf algebras on path algebras of quivers, and more generally on tensor
algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework
of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras
in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two
classification results for actions of semisimple Hopf algebras: the first for
actions which preserve the ascending filtration on tensor algebras, and the
second for actions which preserve the descending filtration on completed tensor
algebras. Extending to more general fusion categories, we illustrate our
classification result for tensor algebras in the pointed fusion categories
${\sf Vec}_{G}^{\omega}$ and in group-theoretical fusion categories, especially
for the representation category of the Kac-Paljutkin Hopf algebra.
