We report for the first time on MBE growth, structural and optical properties of single layers, quantum well structures and short-period superlattices based on BexCd1-xSe ternary alloys on GaAs. Both the conventional MBE growth mode and the sub-monolayer digital alloying technique (SDA) have been employed for the fabrication of the structures. Compositional boundaries of an instability region 0.03 < x < 0.38, calculated in a regular solution approximation for the completely coherent system, agree well with available experimental data. A suppression of the phase separation in BcCdSe by elastic stress in the layer, accompanied by a strong reduction of the Cd incorporation coefficient has been found. Ultrathin 2.8 ML BcCdSe SDA QWs with x similar to 0.15 demonstrate about an order of magnitude increase in the PL intensity with respect to the pure CdSe one, probably resulting from an enhanced carrier localization efficiency. E-g as a function of the Be content reveals a strong bowing in optical data, which allows one to consider BeCdSe alloys with compositions nearly lattice-matched to GaAs as potential materials for the active region of blue-green lasers.