  Backward doubly stochastic Volterra integral equations (BDSVIEs, for short)
are introduced and studied systematically. Well-posedness of BDSVIEs in the
sense of introduced M-solutions is established. A comparison theorem for
BDSVIEs is proved. By virtue of the comparison theorem, we derive the existence
of solutions for BDSVIEs with continuous coefficients. Furthermore, a duality
principle between linear (forward) doubly stochastic Volterra integral equation
(FDSVIE, for short) and BDSVIE is obtained. A Pontryagin type maximum principle
is also established for an optimal control problem of FDSVIEs.
