Kerr microresonators are microscopic ring-shaped cavities that confine light by circulating it in a closed path and enhance the interaction of light through resonance. It has been experimentally observed that the interplay of the Kerr nonlinearity and dispersion in the microresonator can lead to a stable optical signal consisting of a periodic sequence of highly localized ultra-short pulses, resulting in broad frequency spectrum. The discovery that stable broadband frequency combs can be generated in microresonators has unlocked a wide range of promising applications, particularly in optical communications, spectroscopy, and frequency metrology. In its simplest form, the physics in the microresonator is modeled by the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing. In this talk I demonstrate that the Lugiato-Lefever equation indeed supports arbitrarily broad Kerr frequency combs by establishing the existence and stability of periodic solutions consisting of any number of well-separated, strongly localized bright solitons on a single periodicity interval. The existence and spectral stability analyses rely on a rich blend of mathematical tools, such as Lyapunov-Schmidt reduction, Evans-function techniques, bifurcation theory, exponential dichotomies, and high-frequency resolvent bounds. Our analysis confirms that two-mode forcing improves the stability properties of the generated frequency combs compared to one-mode forcing. This is joint work with Lukas Bengel (Karlsruhe Institute of Technology).