We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions d_c. Behaviour at the critical point is non-universal in d>d_c=6 dimensions. Proliferation of the largest clusters, with fractal dimension 4, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are

more »

... eriodic, the maximal clusters have dimension D=2d/3, and obey random-graph asymptotics. Universality is instead manifest at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.