In this research proposal we envisage the study different aspects of multiple scalar field models related with the internal geometry of the field space. Concretely, we consider multi-field models with non-canonical kinetic terms and with non-minimal couplings to gravity. The presence of such couplings induce a non trivial geometric structure in the field space, the field space is generically a curved manifold and its curvature turns out to be crucial for the dynamics of the background and the linear order perturbations of the fields. Additionally, the non geodesic motion in the field space of this models can produce accelerated expansion of the background which is appropriate for modeling the inflationary expansion and the dark energy dominated phases of the evolution of the Universe. The curvature of the field space can lead to different interesting possibilities through the mechanism of “geometric destabilization” consisting in the development of instabilities of the perturbation due to non linear couplings introduced by the field space curvature. During inflation, a negative curvature of the field space can destabilize inflationary trajectories and end inflation abruptly leading to important observational consequences. At the same time, geometric destabilization can be used as a selection and consistency criteria for inflationary models. Geometric destabilization can lead to exponential grow of perturbations which can potentially produce primordial black holes at the mass scales necessary to account for a significative fraction of the present dark matter abundance. It is also expected that this mechanism, joint with the non minimal couplings with gravity could also impact on the reheating dynamics and the dark matter production through novel interactions resulting from the non minimal couplings between the inflaton and dark matter candidates.