The first-principles calculations based on Density Functional Theory (DFT) within generalized gradient approximation (GGA) of Engel-Vosko-Perdew-Wang and modified exact exchange potential of Becke-Johnson have been introduced for the structural and electronic properties of the ScxAl1-xN alloys, respectively. The present lattice constants calculated for the ScAlN alloys and the end compounds (AlN and ScN) are found to be in very good agreement with the available experimental and theoretical ones. The stable ground state structures of the ScxAl1-xN alloys are determined to be wurtzite for the Sc concentration less than similar to 0.403 and rock-salt for the higher Sc concentrations. The present electronic band structure calculations within Becke-Johnson scheme are found to be capable of providing energy band gaps of the AlN and ScN compounds very close to the ones of the available experiments and expensive calculations. According to the calculations of Becke-Johnson potential, the ScxAl1-xN alloys in the wurtzite and zinc-blende structures are direct band gap materials for the Sc concentrations in the ranges of (0.056 <= x <= 0.833) and (0.03125 <= x <= 0.0625, 0.375 <= x <= 0.96875), respectively. However, the ScAlN alloys in the rock-salt phase are determined to be direct band gap materials for total range of the Sc concentration considered in this work. While the energy gaps of the RS-AlScN alloys are found to be extending from near ultraviolet to near infrared with a large (negative) bowing, the ones of the WZ-AlScN and ZB-AlScN alloys are determined to be varying in a small energy range around near ultraviolet with a small (negative) bowing.