Let \(X\subset {\mathbb{R}}^ d\) be a set of positive reach. The curvature measures \(C_ 0(X,\cdot)\),..., \(C_ d(X,\cdot)\) of X were introduced by \textit{H. Federer} [Trans. Am. Math. Soc. 93, 418-491 (1959; Zbl 0089.384)] by a local Steiner formula for the volume of the outer parallel set of X. For smooth X, the \(C_ i(X,\cdot)\) are indefinite integrals of elementary symmetric functions of the principal curvatures of X at boundary points. This representation is generalized by the author to arbitrary sets of positive reach. For this purpose, the curvature measures are expressed as integrals over the normal bundle Nor X of X with respect to the (d-1)-dimensional Hausdorff measure on Nor X. Then, it is shown that the integrands are elementary symmetric functions of principal curvatures which are introduced on Nor X by approximation of X with parallel sets. Finally, the integrals are interpreted as values of associated locally rectifiable currents on appropriately chosen differential forms.