In this talk I will present a mirror symmetry constructions for Jacobians of hyperelliptic curves. In particular, using the abelian/non-abelian correspondence by Ciocan-Fontanine--Kim--Sabbah arXiv:math/0610265, we will calculate its J-function which usually encodes the Gromov-Witten invariants of a variety, but turns out to be nontrivial in this case. Furthermore we will construct a certain family of complex tori that could be considered as the mirror family by some numerical evidences. Based on work "Mirror symmetry for Jacobians of hyperelliptic curves.