In this paper, we introduce and study H-invex functions including the classes of convex, η-invex, (F, G)-invex, c-strongly convex, ϕ-uniformly convex and superquadratic functions, respectively. Each H-invex function attains its global minimum at an H-stationary point. For H-invex functions we prove Jensen, Sherman and Hardy-Littlewood-Pólya-Karamata type inequalities, respectively. We also analyze such inequalities when the control function H is convex. As applications, we give interpretations of the obtained results for uniformly convex and superquadratic functions, respectively.