Local edge parameters on the ASDEX Upgrade tokamak are investigated at the L-mode to H-mode transition, during phases with various types of edge-localized modes (ELMs), and at the density limit. A scaling law for the boundary electron temperature,, is found which describes the H-mode threshold for deuterium-puffed discharges with favourable ion-drift direction. The region of stable operation is bounded by type I ELMs near the ideal ballooning limit and by a minimum temperature necessary to avoid thermal instability of the plasma edge. Stationary operation with type III ELMs imposes an upper limit on the edge temperature. Within the entire range of boundary densities investigated, both L-mode and H-mode are found to be accessible. During type I ELMy H-mode, a relation of global confinement with the edge pressure gradient is found which is connected with a loss of the favourable density dependence predicted by the ITER-92P and ITER-93H ELMy H-mode scalings. At high density, better confinement is achieved in H-modes with an edge pressure gradient below the ideal ballooning limit, eg during type III ELMy H-mode with impurity-seeded radiation.