Partial integrability of almost complex structures

In this talk we will review some old and new results on the problem for local and global existence of holomorphic functions on almost complex manifolds. The following topics will be considered: Existence of holomorphic functions and $IJ$-bundles, Hypersurfaces in $\mathbb{R}^7$, Nearly Kähler manifolds, Homogeneous almost complex spaces, Partial integrability of the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on twistor spaces. Some open problems will also be discussed.