Summary: The paper reviews the main aspects of the renormalization group concepts, from its early applications in quantum field theory to modern asymptotic analysis, and also obtains some new results concerning the quantum quartic oscillator. This problem is approached using the Hamiltonian or flow renormalization methods. After an exposition of several examples, where the Hamiltonian renormalization is applied to exactly solvable models, the flow equations for the quartic oscillator are obtained, and solved numerically, in an approximation suitable to the study of low-lying states. Thus, one finds that the Hamiltonian renormalization is a valuable tool for the study of the quartic oscillator.