We prove an analogue of the classical Steiner formula for the L p affine surface area of a Minkowski outer parallel body for any real parameter p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory as well as the dual quermassintegrals of the dual Brunn Minkowski theory as special cases. Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the s-th mixed L p affine surface area of a Minkowski outer parallel body for any real parameters p and s is also given.