Methods and apparatus for higher-order correction of bandgap voltage references are disclosed. An example bandgap voltage reference circuit disclosed herein comprises a bandgap voltage generation circuit comprising a first resistor, the bandgap voltage generation circuit configured to generate a proportional-to-absolute-temperature current to drive the first resistor to produce a first voltage, the first voltage contributing to an output bandgap voltage, and a first correction circuit electrically coupled to the first resistor and configured to provide a first correction current, the first correction circuit comprising a first nonlinear device configured to generate the first correction current only within a first temperature range, the first correction current decreasing with increasing temperature, the first correction current to drive the first resistor to increase the first voltage only within the first temperature range.