Torsion gravity, or Einstein-Cartan theory, is the simplest
way to go beyond the framework of general relativity. The
main advantage of torsion is the possibility of linking the spin
of fermionic matter to spacetime geometry. Semiclassical
gravity with torsion has been consistently formulated in
80-is, including the renormalizable theory and derivation
of conformal anomaly. After a brief review of the subject,
I will report on the new results concerning the covariant
nonlocal form of the anomaly-induced effective action. This
action manifests a qualitatively new kind of ambiguity coming
from the multiplicative anomaly in the fermionic contribution.
This ambiguity concerns total derivative terms in the
anomaly or, equivalently, the local nonconformal terms in
the effective action. On the other hand, the nonlocal part
of anomaly-induced effective action is free of ambiguity
and admits a low-energy limit in a form similar to the
effective potential in the scalar theory.