The paper presents a quantum theory for the cooling of a cantilever coupled to an optical cavity via radiation pressure. The authors apply the quantum noise approach to the fluctuations of the radiation pressure force, deriving the opto-mechanical cooling rate and the minimum achievable phonon number. They find that reaching the quantum limit of arbitrarily small phonon numbers requires operating in the good cavity regime, where the cavity linewidth is much smaller than the mechanical frequency and the cavity detuning. This regime is in contrast to the common assumption that the mechanical frequency and cavity detuning should be comparable to the cavity damping. The theory is valid for both the good-cavity and bad-cavity regimes, and it provides a basis for interpreting future optomechanical experiments in the quantum regime of mechanical motion.The paper presents a quantum theory for the cooling of a cantilever coupled to an optical cavity via radiation pressure. The authors apply the quantum noise approach to the fluctuations of the radiation pressure force, deriving the opto-mechanical cooling rate and the minimum achievable phonon number. They find that reaching the quantum limit of arbitrarily small phonon numbers requires operating in the good cavity regime, where the cavity linewidth is much smaller than the mechanical frequency and the cavity detuning. This regime is in contrast to the common assumption that the mechanical frequency and cavity detuning should be comparable to the cavity damping. The theory is valid for both the good-cavity and bad-cavity regimes, and it provides a basis for interpreting future optomechanical experiments in the quantum regime of mechanical motion.