Differential operators on almost-Hermitian manifolds and harmonic forms

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.