Cocycles of general form on an arbitrary metric space are considered. The notion of determining modes for cocycles on a Hilbert space is introduced, and a theorem on the existence of finitely many determining modes for such cocycles is proved. The existence of a B-pullback attractor in the problem of microwave heating of a material is proved. The notion of almost periodic integral for a cocycle is introduced, and the existence of such an integral for a certain class of cocycles is proved.