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A theory of particle trapping and transport during passage through a high order one-dimensional nonlinear resonance is developed. It is a dynamic theory of what is generally referred to as the resonance "lock-in" process. The main result is an expression for the trapping efficiency as a function of the resonance excitation width, the nonlinear detuning, and the speed of passage through the resonance. The trapping efficiency shows a characteristic exponential dependence on crossing speed and a dependence on phase*-space area «hich follows from rather general arguments connected with the invariance of local particle density. The question of what phase-space region trapped particles are drawn from is discussed. It is shown how the answer to this question is related to the adiabaticity of the system and the phase-space topology. The phenomenon of particle "explosions", occurring at phase-space amplitudes where nonlinear stabilization becomes ineffective, is investigated. The theory has been verified with a computer simulation for crossing a 5th order resonance.-3-CRISP 74-9 order resonances were sufficiently weak that "fast crossing" was the prevailing mechanism, then the trapping process would substantially decrease in significance, since the trapping quickly diminishes to zero with crossing speed. This, in fact, is probably a good description of the accelerator field before storage rings. With relatively fast tune variations and weak high order resonances in existing accelerators, it is not surprising that the mechanism of trapping, which is enhanced as tune variation diminishes in speed and resonance strength increased, was neglected, if not overlooked. This is not to say that resonances, along with fast crossing, do not cause beam growth. However, this subject has received attention elsewhere and we will not pursue it here.