A process for growing a multielement compound single crystal, includes the steps of placing a crucible holding a raw multielement compound of a predetermined set of composition ratios Y in a vertical crystal growing furnace having a heater, melting the raw multielement compound held in the crucible with the heater to produce a melt of the raw multielement compound in the crucible, controlling the output of the heater to grow a multielement compound single crystal of a predetermined set of composition ratios X from the melt so that the melt is solidified successively upwards from part of the melt in contact with the bottom of the crucible, and feeding to the melt as a solute at least one element of the raw multielement compound from above the level of the melt in the crucible so as to maintain the predetermined set of composition ratios X of the solute during growth of the multielement compound single crystal. The process can keep constant the composition of the grown multielement compound single crystal. The process is applicable to the growth of multielement compound semiconductor single crystals and multielement compound oxide single crystals.