[en] A detailed theoretical analysis is presented of the interaction of intense near-resonant monochromatic radiation with an N-level anharmonic oscillator. In particular, the phenomenon of multiple photon resonance, the process by which an N-level system resonantly absorbs two or more photons simultaneously, is investigated. Starting from the Schroedinger equation, diagrammatic techniques are developed that allow the resonant process to be analyzed quantitatively, in analogy with well-known two-level coherent phenomena. In addition, multiple photon Stark shifts of the resonances, shifts absent in two-level theory, are obtained from the diagrams. Insights into the nature of multiple photon resonances are gained by comparing the quantum mechanical system with classical coupled pendulums whose equations of motion possess identical eigenvalues and eigenvectors. In certain limiting cases, including that of the resonantly excited N-level harmonic oscillator and that of the equally spaced N-level system with equal matrix elements, analytic results are derived. The influence of population relaxation and phase-disrupting collisions on the multiple photon process are also analyzed, the latter by extension of the diagrammatic technique to the density matrix equations of motion. 11 figures