  For backward stochastic Volterra integral equations (BSVIEs) in
multi-dimensional Euclidean spaces, comparison theorems are established in a
systematic way for the adapted solutions and adapted M-solutions. For
completeness, comparison theorems for (forward) stochastic differential
equations, backward stochastic differential equations, and (forward) stochastic
Volterra integral equations (FSVIEs) are also presented. Duality principles are
used in some relevant proofs. Also, it is found that certain kind of
monotonicity conditions play crucial roles to guarantee the comparison theorems
for FSVIEs and BSVIEs to be true. Various counterexamples show that the assumed
conditions are almost necessary in some sense.
