Nonautonomous bifurcation theory studies the change of attractors of nonautonomous systems which are introduced here with the process formalism as well as the skew product formalism. We present a total stability theorem ensuring the existence of nearby attractors of perturbed systems. They depend continuously on a parameter if and only if the attraction is uniform w.r.t. parameter, i.e. the attractors are equiattracting. We apply these principles to explicit systems to clarify the meaning of continuous and abrupt transitions of attractors in contrast to bifurcations, i.e. splitting of minimal invariant subsets into others within the attractor. Several examples are treated, including a nonautonomous pitchfork bifurcation.