  In this paper we consider the effects of adding curvature in extended
cosmologies involving a free-to-vary neutrino sector and different
parametrizations of Dark Energy (DE). We make use of the Planck 2018 cosmic
microwave background temperature and polarization data, Baryon Acoustic
Oscillations and Pantheon type Ia Supernovae data. Our main result is that a
non-flat Universe cannot be discarded in light of the current astronomical
data, because we find an indication for a closed Universe in most of the DE
cosmologies explored in this work. On the other hand, forcing the Universe to
be flat can significantly bias the constraints on the equation of state of the
DE component and its dynamical nature.
