In this article, a class of backward stochastic Volterra integro-differential equations (BSVIDEs) is introduced and studied. It is worthy mentioning that the proposed BSVIDEs can not be covered by the existing backward stochastic Volterra integral equations (BSVIEs), and they also have the nice flow property such that Ito's formula becomes quite applicable. It is found that BSVIDEs can provide a neat sufficient condition for the solvability of BSVIEs with generator depending on the diagonal value of the solutions. As applications, the optimal control problems in terms of maximum principles and linear quadratic control problems of optimal control for forward stochastic Volterra integro-differential equations (FSVIDE) are investigated. In contrast with the BSVIEs in current literature, some interesting phenomena and advantages of BSVIDEs are revealed.