This review is concerned with applying the analytic signal in oscillation theory, where the concept of the analytic signal was hardly ever applied until recently. We treat the mathematical properties of the Hilbert transform and of the analytic signal, which allow one to determine the amplitude, phase, and frequency of any oscillation at any instant of time. For narrow-band oscillations and for broad-band oscillations that arise under slow frequency modulation, this definition agrees with the intuitive meaning of amplitude, phase, and frequency and with the quasistationary approximation, while allowing one to estimate the limits of applicability of the latter. We show that a number of radiotechnical devices (mixers, frequency modulators, detectors, frequency discriminators, etc.) transform the parameters of an oscillation as defined by the analytic signal. We establish the relationship between the adiabatic invariant and the equation of the oscillations for the analytic signal. This relationship allows one to construct a complete theory of the triode oscillator having a cubic characteristic, in which the capacity of the circuit and the transconductance of the tube slowly fluctuate. Here we get a new result in the second approximation, namely: we calculate the influence of the flicker effect on the instantaneous frequency of the oscillator; the corresponding spectral line width is substantial in practice. In conclusion, we treat some paradoxes and supplementary examples that illustrate the technical and physical significance of the introduced concepts.