A OLED display and a driving method thereof are disclosed. The OLED display includes: an OLED display panel including: data lines to which data voltages are supplied; gate lines to which a gate voltage is sequentially supplied; luminescence control lines to which a luminescence control voltage is sequentially supplied, a driving power line to which a driving voltage is supplied; a compensation power line to which a compensation voltage having a first level and a second level different from the first level are supplied; a plurality of pixel cells each respectively in pixel areas defined by the data lines and the gate lines; a data driver having output lines whose number is smaller than the number of the data lines; and a demultiplexer unit formed between the data driver and the OLED display panel, the demultiplexer unit supplying the data voltages from the output lines to the data lines, wherein each of the pixel cells includes: a light emitting element; and a pixel driver that supplies a current corresponding to a corresponding one of the data voltages to the light emitting element based on the corresponding data voltage, the gate voltage, the luminescence control voltage, the driving voltage and the compensation voltage having the first level and that turns off the light emitting element when the compensation voltage has the second level.