In this paper, we present several new properties of h-convex functions in a way that complements those known properties for convex functions. The obtained results include, but are not limited to, Mercer-type inequalities, gradient inequalities, Jensen-type inequalities, Mean-like bounds, Hermite-Hadamard inequalities, external behavior, and super-additive inequalities. The obtained results, then, are employed to obtain some applications related to matrix inequalities, including unital positive mappings, weak majorization, and trace inequalities that generalize the celebrated Klein inequality.