We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper [arXiv:1110.0605], where we introduced the concepts of locally class-presentable and class-accessible categories. In this work we prove that the category of weak equivalences of a nice class-combinatorial model category is class-accessible. Our extension of J. Smith localization theorem depends on the verification of a cosolution-set condition. The deepest result is that the (left Bousfield) localization of a class-combinatorial model category with respect to a strongly class-accessible localization functor is class-combinatorial again.