|This volume of the EMS is devoted to applications of singularity theory in mathematics and physics. The authors Arnol'd, Vasil'ev, Goryunov and Lyashko study bifurcation sets arising in various contexts such as the stability of singular points of dynamical systems, boundaries of the domains of ellipticity and hyperbolicity of partial differentail equations, boundaries of spaces of oscillating linear equations with variable coefficients and boundaries of fundamental systems of solutions. The book also treats applications of the following topics: functions on manifolds with boundary, projections of complete intersections, caustics, wave fronts, evolvents, maximum functions, shock waves, Petrovskij lacunas and generalizations of Newton's topological proof that Abelian integrals are transcendental. The book contains descriptions of numberous very recent research results that have not yet appeared in monograph form. There are also sections listing open problems, conjectures and directions of future research. It will be of great interest for mathematicians and physicists, who use singularity theory as a reference and research aid.