In this article, we use quantum Langevin equations to provide a theoretical understanding of the non-classical behavior of Kerr optical frequency combs when pumped below and above threshold. In the configuration where the system is under threshold, the pump field is the unique oscillating mode inside the resonator, and triggers the phenomenon of spontaneous four-wave mixing, where two photons from the pump are symmetrically up- and down-converted in the Fourier domain. This phenomenon can only be understood and analyzed from a fully quantum perspective as a consequence of the coupling between the field of the central (pumped) mode and the vacuum fluctuations of the various sidemodes. We analytically calculate the power spectra of the spontaneous emission noise, and we show that these spectra can be either single- or double peaked depending on the parameters of the system. We also calculate as well the overall spontaneous noise power per sidemode, and propose simplified analytical expressions for some particular cases. In the configuration where the system is pumped above threshold, we investigate the phenomena of quantum correlations and multimode squeezed states of light that can occur in the Kerr frequency combs originating from stimulated four-wave mixing. We show that for all stationary spatio-temporal patterns, the side-modes that are symmetrical relatively to the pumped mode in the frequency domain display quantum correlations that can lead to squeezed states of light. We also explicitly determine the phase quadratures leading to photon entanglement, and analytically calculate their quantum noise spectra. We finally discuss the relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths, below and above threshold.
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