Abstract: Our existence is basically linked to the fact that matter dominates over antimatter in the Universe. Several approaches in modern Physics exist, which provide explanations on the above fact. In conventional frameworks, where Lorentz and CPT invariance is assumed to describe the cosmological evolution, the celebrated Sakharov’s conditions, namely violations of charge conjugation C, charge-parity CP, and Baryon number B, as well as departure from equilibrium, are required for this purpose. All such requirements are met qualitatively, but not quantitatively, within the framework of the standard model of particle physics. In this colloquium, I will discuss the possibility of a new paradigm shift in the understanding of the matter-antimatter asymmetry in the Cosmos, which is based on cosmologies inspired from string theory. The latter are characterised by the presence of (quantum) gravitational anomalies in the very early Universe, coupled to an axion field that characterise the massless gravitational string multiplet. The low-energy gravitational theory in such a case is a Chern-Simons gravity. The anomaly terms are non-trivial in the presence of chiral (that is, left-right asymmetric) gravitational waves. Chiral primordial gravitational waves in such cosmologies can lead to a condensation of the gravitational anomaly terms. Estimates of such condensates can be provided within the framework of weak (perturbative) quantum gravity. This, in turn, leads to a metastable inflationary phase, of the so-called running-vacuum type, but also to a specific background configuration for the axion, characterised by a constant cosmic rate, which breaks Lorentz and CPT symmetry spontaneously. Such axion fields remain undiluted until the exit from inflation, and onto the radiation era, and can lead to matter-antimatter asymmetries in such universes in an unconventional way. In the talk, I will explain the basic features of the scenario, and argue how the model can be compatible with general covariance, and also consistent with the plethora of the current cosmological data, including the possibility of providing an alleviation of the present-epoch Hubble and growth-of-structure tensions. In this last respect, I will compare this string-inspired model with other running-vacuum cosmologies.