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