  I review several aspects of torsion in string-inspired cosmologies. In
particular, I discuss its connection with fundamental, string-model
independent, axion fields associated with the massless gravitational multiplet
of the string. I argue in favour of the role of primordial gravitational
anomalies coupled to such axions in inducing inflation of a type encountered in
the "running-vacuum-model (RVM)" cosmological framework, without fundamental
inflaton fields. The gravitational-anomaly terms owe their existence to the
Green-Schwarz mechanism for the (extra-dimensional) anomaly cancellation, and
may be non-trivial in such theories in the presence of (primordial)
gravitational waves at early stages of the four-dimensional string Universe
(after compactification). I also discuss how the torsion-induced stringy axions
can acquire a mass in the post inflationary era, due to non perturbative
effects, thus having the potential to play the r\^ole of (a component of) dark
matter in such models. Finally, I describe briefly the current-era
phenomenology of this model, with the emphasis placed on the possibility of
alleviating tensions observed in the current-era cosmological data. A brief
phenomenological comparison with other cosmological models in contorted
geometries is also made.
