In this talk we will focus on the derivation and the analysis of the inhomogeneous kinetic wave equation (IKWE). Although there is an extended amount of literature for the space homogeneous KWE, the inhomogeneous problem has barely been addressed. In the first part we examine the validity of the IKWE for quadratic nonlinearities and dispersion relations close to Laplacian, based on joint work with Collot, Germain. In the second part of the talk, we investigate the global well-posedness and stability of the 4-wave inhomogeneous kinetic wave equation near vacuum, employing techniques from classical kinetic theory, such as the Kaniel-Shinbrot iteration.