The automorphism groups scheme of an algebraic variety is a fundamental invariant of an algebraic variety and its structure is intimately related to its moduli theory. In this talk I will present some results about the structure of the automorphism group scheme of a minimal surface of general type defined over an algebraically closed field of positive characteristic. In particular I will emphasize on certain differences between the characteristic zero case and the positive characteristic case, like the existence of surfaces of general type with non reduced automorphism group scheme, a situation that appears exclusively in positive characteristic.