In this talk, a backward stochastic Volterra integral equation (BSIE, for short) is introduced. Mathematically, this is a natural extension of backward stochastic differential equation. It turns out that BSIEs are closely related to stochastic models with memories, in particular, the so-called time-inconsistent preferences, nonlinear expectations, dynamic risk measures, stochastic differential utilities, etc. We will present well-posedness of BSIEs. Various properties for the adapted solutions to BSIEs will be established, such as duality principle between linear BSIEs and (forward) stochastic Volterra integral equations, comparison theorem, etc. Also, a Pontryagin type maximum principle for optimal control of stochastic integral equations will be presented.