The principal models and phenomena of the modem theory of nonlinear oscillations and waves are briefly reviewed in observance of L. I. Mandel'shtam's centennial. Each part of the paper, broken down into sections on ``Oscillators'', ``Self-oscillations'', and ``Modulation'', begins with a discussion of the classical models and effects of the role they play in the modern theory. Attention is concentrated on an analysis of complex behavior of simple systems, collective phenomena and ensembles of nonlinear oscillators, the initiation of self-oscillations in space, stochastic self-oscillations and waves, and the excitation, transfer, and origins of modulation in nonlinear systems and media, etc. Oscillation-theory and wave-theory models are discussed in parallel.