The cyclotron resonance laser (CRL) accelerator is a novel concept of accelerating continuous charged-particle beams to moderately or highly relativistic energies. This paper discusses prospects and limitations of this concept. In particular, the nonlinear coupling of an intense traveling electromagnetic wave with an electron beam in a guide magnetic field is studied, and the effects of wave dispersion on particle acceleration are analyzed. For a tenuous beam, it is shown in a single-particle theory that the maximum energy gain and the maximum acceleration distance for the beam electrons in CRL accelerators with optimal magnetic taper exhibit power-law scaling on the degree of wave dispersion (measured by the parameter w/ckjj -1). The maximum energy gain is found to be independent of the wave amplitude, and the accelerating gradient is proportional to the wave amplitude. A self-consistent multiparticle model, which confirms the validity of the scaling laws, is used to study the characteristics of CRL accelerators in moderately high-current, microwave regimes. The parameter regimes of experimental interest are identified. The possibility of building continuous-wave (CW) CRL accelerators is discussed. Finally, the results of simulation modelling of a multi-megavolt, X-band, electron CRL accelerator are presented.