Amplitude-modulated cross-polarization (AMCP) schemes that can greatly improve the transfer efficiency under fast magic angle spinning am discussed. A novel class of pulse sequences (S-AMCP) that employ an amplitude-modulated spin-lock field on the S-spin channel only and cw irradiation on the I-spin channel is introduced. It leads to efficient cross polarization at the standard Hartmann-Hahn condition while maintaining good spin-lock properties. S-AMCP does, however, not decrease the sensitivity to exact Hartmann-Hahn matching that can become critical under fast rotation. A double amplitude modulation scheme (D-AMCP) that uses amplitude modulation on both radiofrequency channels and significantly broadens the Hartmann-Hahn matching condition is presented. It can, however, be applied only to spin systems with relatively low I-spin homonuclear dipolar interactions due to the absence of an effective spin-locking field. S-AMCP pulse sequences, in contrast, are generally applicable and yield cross-polarization efficiencies that are always better than or equal to the standard continuous-wave cross-polarization experiment.