In this talk we first recall the development on the study of the (equivariant) unitary bordism of unitary manifolds, and list some problems. We show that the equivariant unitary bordism is determined by ordinary equivariant cohomology Chern numbers, which answers the conjecture posed by Guillemin--Ginzburg—Karshon. One of our approaches is the use of the toric genus and the Kronecker pairing of bordism and cobordism; another approach is that we employ the method of tom Dieck, which has successfully been used to deal with the case of equivariant unoriented bordism. In addition, we also answer the Buchstaber–Panov–Ray Problem in 2010 as follows: “for any set of complex $T^k$-representations $W_x$, is there a necessary and sufficient conditions for the existence of a tangentially stably complex $T^k$-manifold with the given representations as fixed point data?” This talk is based on the joint work with Jun Ma and Wei Wang