The present book carefully studies the blow-up phenomenon of solutions to partial differential equations, including many equations of mathematical physics. The included material is based on lectures read by the authors at the Lomonosov Moscow State University, and the book is addressed to a wide range of researchers and graduate students working in nonlinear partial differential equations, nonlinear functional analysis, and mathematical physics.ContentsNonlinear capacity method of S. I. PokhozhaevMethod of self-similar solutions of V. A. GalaktionovMethod of test functions in combination with method of nonlinear capacityEnergy method of H. A. LevineEnergy method of G. TodorovaEnergy method of S. I. PokhozhaevEnergy method of V. K. Kalantarov and O. A. LadyzhenskayaEnergy method of M. O. Korpusov and A. G. SveshnikovNonlinear Schr?dinger equationVariational method of L. E. Payne and D. H. SattingerBreaking of solutions of wave equationsAuxiliary and additional results