The GENERIC structure of mesoscopic time evolution equations is introduced as a natural extension of mesoscopic equilibrium thermodynamics. Equilibrium theory on any level of description is proposed to be represented by the maximum entropy principle or, in a more mathematical language, by the dominant role that the Legendre transformations play in its formulation. The equilibrium theory is extended to a non-equilibrium theory by keeping the dominant role for this type of transformations. The passage from a mesoscopic equilibrium theory to the classical equilibrium thermodynamics (a Legendre transformation in the context of equilibrium theories) is required to be made as a continuous sequence of Legendre transformations representing the time evolution. We show that the equations generating such time evolution possess the GENERIC structure. (C) 2010 Elsevier B.V. All rights reserved.