I shall discuss the construction of Hermitian Yang-Mills instantons over Calabi-Yau cones, as well as over their resolutions. In particular, in d complex dimensions, I shall present an infinite family of $SU(d)$ instantons, parametrised by an integer $k$ and a continuous modulus. A detailed study of their properties, including the computation of the instanton numbers is provided. Then, I shall also explain how they can be used in the construction of heterotic non-Kahler solutions, including heterotic string duals of 4d $N=1$ SYM theories.