The study of nonlocal bifurcations from the topological point of view requires not only topological, but smooth normal forms of the families of differential equations near singular points. In the first part of the paper a survey of these normal forms is presented. In the second part these normal forms are applied to the study of the bifurcations of planar vector fields. A complete list of polycycles appearing in generic two or three parameter families (Zoo of Kotova) is presented. The proof of the finite cyclicity of elementary polycycles occuring in typical finite parameter families of planar vector fields is outlined.