The book describes how curvature measures can be introduced for certain classes of sets with
singularities in Euclidean spaces. Its focus lies on sets with positive reach and some
extensions which include the classical polyconvex sets and piecewise smooth submanifolds as
special cases. The measures under consideration form a complete system of certain Euclidean
invariants. Techniques of geometric measure theory in particular rectifiable currents are
applied and some important integral-geometric formulas are derived. Moreover an approach to
curvatures for a class of fractals is presented which uses approximation by the rescaled
curvature measures of small neighborhoods. The book collects results published during the last
few decades in a nearly comprehensive way.
