  Oeljeklaus-Toma (OT) manifolds are complex non-K\"ahler manifolds whose
construction arises from specific number fields. In this note, we compute their
de Rham cohomology in terms of invariants associated to the background number
field. This is done by two distinct approaches, one using invariant cohomology
and the other one using the Leray-Serre spectral sequence. In addition, we
compute also their Morse-Novikov cohomology. As an application, we show that
the low degree Chern classes of any complex vector bundle on an OT manifold
vanish in the real cohomology. Other applications concern the OT manifolds
admitting locally conformally K\"ahler (LCK) metrics: we show that there is
only one possible Lee class of an LCK metric, and we determine all the possible
Morse-Novikov classes of an LCK metric, which implies the nondegeneracy of
certain Lefschetz maps in cohomology.
