H-modes exhibiting improved confinement above the L-mode are achieved in Alcator C-Mod with ICRF and with ohmic heating alone without boronization. Both ELM-free and ELMy H-modes are obtained with total input power from 0.75 to 4.2 MW over a range of densities (0.8 to ) and toroidal fields (3 to 8 T). Type III ELMs are often observed to have coherent, high m and n precursor oscillations with frequencies of 100 - 160 kHz. The threshold power required to achieve the H-mode increases with density and toroidal field, in rough agreement with scalings derived from other tokamaks. The power densities and density times toroidal field products are an order of magnitude larger than in other tokamaks, in the range of values expected for ITER. The L - H and H - L transitions occur at approximately the same edge electron temperature. A low density limit to the H-mode is found at about . A high midplane neutral pressure limit of about 0.6 mTorr is also observed.