The Gibbsian relation is of fundamental importance to the thermodynamics of nonequilibrium systems. In this paper, we shall present an analytical derivation and several generalizations of this relation for dilute, nonequilibrium and certain highly nonequilibrium, systems. Our analysis will beindependent of the collision dynamics, because it will be based on the general kinetic equation witharbitrary collision integrals. Consequently, our analysis can provide athermodynamic derivation and several generalizations of the Gibbsian relation. Our distribution functions can also admit some arbitrary, nonequilibrium and highly nonequilibrium, forms. With the help of the generalized Gibbsian relation and a fundamental axiom to be postulated, the entropy production rates and the generalized forces and fluxes will be studied for our highly nonequilibrium systems. The second law of thermodynamics will be postulated and verified in specific cases. Onsager's reciprocity relations will be discussed.