Reactive-ion molecular-beam epitaxy has been used to grow epitaxial hexagonal-structure alpha-GaN on Al2O3(0001) and Al2O3(0112BAR) substrates and metastable zinc-blende-structure beta-GaN on MgO(001) under the following conditions: growth temperature T(s) = 450-800-degrees-C; incident N2+/Ga flux ratio J(N2+)/J(Ga) = 1-5; and N2+ kinetic energy E(N2+) = 35-90 eV. The surface structure of the alpha-GaN films was (1 X 1), with an almost-equal-to 3% contract ion in the in-plane lattice constant for films grown on Al2O3(0001), while the beta-GaN films exhibited a 90-degrees-rotated two-domain (4 X 1) reconstruction. Using a combination of in situ reflection high-energy electron diffraction, double-crystal x-ray diffraction, and cross-sectional transmission electron microscopy, the film/substrate epitaxial relationships were determined to be: (0001)GaN parallel-to (0001)Al2O3 with [2110BAR]GaN parallel-to [1100BAR]Al2O3 and [1100BAR]GaN parallel-to [1210BAR]Al2O3, (2110BAR)GaN parallel-to (0112BAR)Al2O3 with [0001]GaN parallel-to [0111BAR]Al2O3 and [0110BAR]GaN parallel-to [2110BAR]A12O3, and (001)GaN parallel-to (001)MgO with [001]GaN parallel-to [001]MgO. Films with the lowest extended defect number densities (n(d) congruent-to 10(10) cm-2 threading dislocations with Burgers vector a0/3[1120BAR]) and the smallest x-ray-diffraction omega rocking curve widths (5 min) were obtained using Al2O3(0001) substrates, T(s) greater-than-or-equal-to 650-degrees-C, J(N2+)/J(Ga) greater-than-or-equal-to 3.5, and E(N2+) = 35 eV. Higher N2+ acceleration energies during deposition resulted in increased residual defect densities. In addition, E(N2+) and J(N2+)/J(Ga) were found to have a strong effect on film growth kinetics through a competition between collisionally induced dissociative chemisorption of N2 and stimulated desorption of Ga as described by a simple kinetic growth model. The room-temperature resistivity of as-deposited GaN films grown at T(s) = 600-700-degrees-C with E(N2+) = 35 eV increased by seven orders of magnitude, from 10(-1) to 10(6) OMEGA cm, with an increase in J(N2+)/J(Ga) from 1.7 to 5.0. Hall measurements on the more conductive samples yielded typical electron carrier concentrations of 2 X 10(18) cm-3 with mobilities of 30-40 cm2 V-1 s-1. The room-temperature optical band gaps of alpha-GaN and beta-GaN were 3.41 and 3.21 eV, respectively.