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In this paper we introduce several new cohomologies of almost complexmanifolds, among which stands a generalization of Bott-Chern and Aepplicohomologies defined using the operators $d$, $d^c$. We explain how they areconnected to already existing cohomologies of almost complex manifolds and westudy the spaces of harmonic forms associated to $d$, $d^c$, showing theirrelation with Bott-Chern and Aeppli cohomologies and to other well-studiedspaces of harmonic forms. Notably, Bott-Chern cohomology of $1$-forms isfinite-dimensional on compact manifolds and provides an almost complexinvariant $h^1_{d + d^c}$ that distinguishes between almost complex structures.On almost K\"ahler $4$-manifolds, the spaces of harmonic forms we consider areparticularly well-behaved and are linked to harmonic forms considered by Tsengand Yau in the study of symplectic cohomology.

On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms