Three authors, from Germany, Russia, and Belorussia, examine the two related notions. They describe partial integral operators as linear or nonlinear operators with integrals that depend on a parameter. The integro-differential equations are of the Barbashin type in that the right-hand sides contain both a multiplication operator and an integral operator. The connection, they explain, is that an initial value problem for a Barbashin integro-differential equation, linear or nonlinear, leads in a natural way to a fixed-point problem involving a partial integral operator. Drawing on research in Russia much of which has not even been referenced in the west, they cover first the equations of the Barbashin type and the theory behind them, then the operators. They close with generalizations and applications of each.