The deformed Hermitian--Yang--Mills equation, which will be abbreviated as dHYM equation, was discovered around the same time by Mariño--Minasian--Moore--Strominger and Leung--Yau--Zaslow using different points of view. Mariño--Minasian--Moore--Strominger found out that the dHYM equation is the requirement for a -brane on the -model of Mirror Symmetry to be supersymmetric. It was shown by Leung--Yau--Zaslow that, in the semi-flat model of Mirror Symmetry, solutions of the dHYM equation correspond to a Fourier--Mukai transform to special Lagrangian submanifolds of the Mirror. The study of the dHYM equation for a holomorphic line bundle over a compact Kähler manifold was initiated by Jacob and Yau. In this talk, we will extend the results on compact Hermitian manifolds and will show some interesting applications.