Variations in the Sun's total energy output (luminosity) are caused by changing dark (sunspot) and bright structures on the solar disk during the 11-year sunspot cycle. The variations measured from spacecraft since 1978 are too small to have contributed appreciably to accelerated global warming over the past 30 years. In this Review, we show that detailed analysis of these small output variations has greatly advanced our understanding of solar luminosity change, and this new understanding indicates that brightening of the Sun is unlikely to have had a significant influence on global warming since the seventeenth century. Additional climate forcing by changes in the Sun's output of ultraviolet light, and of magnetized plasmas, cannot be ruled out. The suggested mechanisms are, however, too complex to evaluate meaningfully at present.