to introduce new twistor actions for bosonic and ${{\cal N}}{=}4$ supersymmetric Yang-Mills theories.

Specializing to the cases $M^4=S^4$, ${\mathbb C}P^2$, $B_4$ and ${\mathbb C}B_2$, we describe Kähler, nearly Kähler and nearly Calabi-Yau structures on the twistor spaces ${{\cal T}}(M^4)$ for these four cases. Various explicit solutions of the Hermitian-Yang-Mills equations on ${{\cal T}}(M^4)$ will be written down and their applications in flux compactifications of heterotic strings will be briefly discussed.

Pseudo-holomorphic bundles and Hermitian-Yang-Mills equations
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[**Notation**]{}. Let $X^{2n}$ be an oriented Riemannian $2n$-dimensional manifold and $\{e^a\}$ a local orthonormal basis of $T^*X^{2n}$ , $a=1,...,2n$. For $p$-forms on $X^{2n}$ we use the notation $$\label{2.1}
F_p=\sfrac{1}{p!}\,F_{a_1...a_p}\, e^{a_1...a_p}{{\qquad{\rm with}\qquad}}e^{a_1...a_p}:=e^{a_1}\wedge ... \wedge e^{a_p}\ ,$$ $$\label{2.2}
(*F_p)_{a_1...a_{2n-p}}=\sfrac{1}{p!}\,\ve_{a_1...a_{2n-p}\, b_1...b_p}\,
F^{b_1...b_p}\quad\Leftrightarrow\quad
F_p\wedge *F_p= \sfrac{1}{p!}\, F_{a_1...a_p}\,F^{a_1...a_p}\,{vol}_{2n} \ ,$$ where $*$ is the Hodge star operator and ${vol}_{2n}=e^1\wedge...\wedge e^{2n}$. We also use notation [@CS] $$\label{2.3}
(F_p\lrc\, S_{p+q})_{b_1...b_q}=\sfrac{1}{p!}\, (F_p)^{a_1...a_p}
(S_{p+q})_{a_1...a_p\,b_1...b_q}$$ that exploits the underlying Riemannian metric $\diff s^2 = \de_{ab}\, e^ae^b$ with the convention that $e^{12}\lrc\, e^{123}=e^3$ etc.

[**Pseudo-holomorphic bundles**]{}. Consider an oriented $2n$-dimensional manifold $X^{2n}$ with an almost complex structure ${{\cal J}}$ and a complex vector bundle ${{\cal E}}$ over $X^{2n}$ endowed with a connection ${{\cal A}}$. According to R.Bryant [@Bryant], a connection ${{\cal A}}$ on ${{\cal E}}$ 