In this talk, we will study the dissipative structure of both incompressible and compressible MHD systems. First, we focus on the 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. We also focus on the 2D incompressible MHD system with only magnetic diffusion on periodic domain. Then we focus on the 2D compressible viscous and non-resistive MHD equations. We derive the global existence of small smooth solutions to this system on periodic domain.