High voltage GaN Schottky and p-i-n rectifiers have been fabricated on heteroepitaxial layers. The Schottky diodes have reverse blocking voltages around 500 V for vertical devices employing undoped, conducting GaN, whereas these voltages are >3000 V for lateral devices employing resistive GaN, The forward turn-on voltages are greater than or equal to 3.5 V for the Schottky rectifiers, with ideality factors of 1.5-2. The dominant current transport mechanism is Shockley-Read-Hall recombination. The p-i-n rectifiers fabricated with conducting i layers also have reverse blocking voltages of similar to 500 V, but the forward turn-on voltages are typically similar to 5 V. A comparison with the state-of-the-art SiC Schottky and p-i-n rectifiers is also given.