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We prove the boundedness of the maximal operator in generalized Orlicz spaces defined on subsets of R n. The proof is based on an extension result for Φ-functions. We study generalized Sobolev–Orlicz spaces and establish density of smooth functions and the Poincaré inequality. As applications we establish the existence of solutions of the φ-Laplace equation with zero and non-zero right-hand side. Further, we systematize assumptions for Φ-functions and prove several basic tools needed for the study of differential equations of generalized Orlicz growth.