  We study cohomologies on an almost complex manifold $(M, J)$, defined using
the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from
the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as
vector-valued forms on $M$. We show how one of these, the $N$-cohomology
$H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable
almost complex structures on $M$. Another one, the $J$-cohomology
$H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its
definition and applicability to the case of non-integrable almost complex
structures. The $J$-cohomology encodes whether a complex manifold satisfies the
$\partial \bar{\partial}$-lemma, and more generally in the non-integrable case
the $J$-cohomology encodes whether $(M, J)$ satisfies the $\mathrm{d}
\mathcal{L}_J$-lemma, which we introduce and motivate in this paper. We discuss
several explicit examples in detail, including a non-integrable example. We
also show that $H^k_J$ is finite-dimensional for compact integrable $(M, J)$,
and use spectral sequences to establish partial results on the
finite-dimensionality of $H^k_J$ in the compact non-integrable case.
