  We review topological properties of K\"ahler and symplectic manifolds, and of
their odd-dimensional counterparts, coK\"ahler and cosymplectic manifolds. We
focus on formality, Lefschetz property and parity of Betti numbers, also
distinguishing the simply-connected case (in the K\"ahler/symplectic situation)
and the $b_1=1$ case (in the coK\"ahler/cosymplectic situation).
