In this paper we study the relation between the tangent structure of an analytic set V at a point p and the local representation of V as a branched covering. A prototype for our type of result is the fact that one obtains a covering of minimal degree by projecting transverse to the Zariski tangent cone C 3 ( V , p ) {C_3}(V,p) . We show, for instance, that one obtains the smallest possible branch locus for a branched covering if one projects transverse to the cone C 4 ( V , p ) {C_4}(V,p) . This and similar results show that points where the various tangent cones C i ( V , p ) , i = 4 , 5 , 6 {C_i}(V,p),i = 4,5,6 , have minimal dimension give rise to the simplest branched coverings. This observation leads to the idea of “Puiseux series normalization", generalizing the situation in one dimension. These Puiseux series allow us to strengthen some results of Hironaka and Whitney on the local structure of certain types of singularities.