New methods and approaches of the global bifurcation analysis of non-linear dynamical systems, described by nonlinear ODE or difference equations, are under consideration. The main idea of new approaches is a concept of com-plete bifurcation groups, periodic skeletons, protuberances and rare attractors. The method of complete bifurcation groups (MCBG) allows to obtain new qualitative results in well known dynamical models. New results obtained by the method are discussed in this paper for the typical driven oscillatory and vibro-impact systems with one and several degrees-of-freedom. The concept of rare attractors and pro-tuberances illustrated by typical bifurcation groups allows to find new important applications. Obtained results are important for explanations of some catastrophic or good-luck events in engineering.