Central values of $L$-functions have been investigated extensively in the literature due to significant arithmetic information carried by them. Moments of central values of $L$-functions play important roles in our understanding ofthe non-vanishing problem of central values of $L$-funtions. In this talk, I will present some results on the moments of central values of cubic Hecke $L$-functions of the Gaussian field, which in turn imply a quantitative non-vanishing result for the $L$-values. This is joint work with L. Zhao.