  Using cohomological methods, we identify both trivial and nontrivial
contributions to the conformal anomaly in the presence of vectorial torsion in
$d=2,4$ dimensions. In both cases, our analysis considers two scenarios: one in
which the torsion vector transforms in an affine way, i.e., it is a gauge
potential for Weyl transformations, and the other in which it is invariant
under the Weyl group. An important outcome for the former case in both $d=2,4$
is the presence of anomalies of a "mixed" nature in relation to the
classification of Deser and Schwimmer. For invariant torsion in $d=4$, we also
find a new type of anomaly which we dub $\Psi$-anomaly. Taking these results
into account, we integrate the different anomalies to obtain renormalized
anomalous effective actions. Thereafter, we recast such actions in the
covariant nonlocal and local forms, the latter being easier to work with. Along
the way, we pause to comment on the physical usefulness of these effective
actions, in particular to obtain renormalized energy-momentum tensors and
thermodynamics of $2d$ black holes.
