A voltage regulator and a power supply are provided. The voltage regulator includes an operational amplifier and an offset voltage control module. The offset voltage control module includes one or more stages of regulation branches connected in parallel, and controls an offset voltage of the operational amplifier with the one or more stages of regulation branches to regulate the output voltage. The offset voltage control module also includes a bandgap reference generation circuit, configured to generate a reference voltage irrelevant to a temperature coefficient that is received by the operational amplifier from the input terminal, wherein the bandgap reference generation circuit comprises at least one of: a ΔVGS-based bandgap reference generation circuit having a full CMOS reference offset structure, a PTAT unit-based and VGS-based bandgap reference generation circuit having a full CMOS reference offset structure, and a PTAT unit-based and BJT-based bandgap reference generation circuit having a complementary structure.