  We study the resonant enhancement of linear and nonlinear chiroptical effects
in asymmetric silicon metasurfaces supporting multipolar Mie resonances and
quasi-bound states in the continuum (quasi-BICs). We demonstrate theoretically
and observe in experiment the pronounced linear circular dichroism at the
quasi-BIC resonances. We further find that both local field enhancement and
third-harmonic signal are large for Mie resonances and some quasi-BIC modes. We
explain the selectivity of the nonlinear enhancement by employing the concept
of critical coupling being more favorable for the modes with moderately large
radiative quality factors ($Q$ factors). We demonstrate experimentally strong
nonlinear chiroptical response associated with high efficiency of the
third-harmonic generation and large nonlinear circular dichroism varying from
$+0.918\pm0.049$ to $-0.771\pm0.004$ for the samples with different
asymmetries. We believe our results suggest a general strategy for engineering
nonlinear chiroptical response in dielectric resonant metasurfaces.
