We analyse electron and hole transport in organic light-emitting diodes (OLEDs) via the drift-diffusion equations. We focus on space-charge-limited transport, in which rapid variations in charge carrier density occur near the injecting electrodes, and in which the electric field is highly non-uniform. This motivates our application of singular asymptotic analysis to the drift-diffusion equations. In the absence of electronhole recombination, our analysis reveals three regions within the OLED: (i) 'space-charge layers' near each electrode whose widthλ s is much smaller than the device widthL, wherein carrier densities decay rapidly and the electric field is intense; (ii) a 'bulk' region whose width is on the scale ofL, where carrier densities are small; and (iii) intermediate regions bridging (i) and (ii). Our analysis shows that the currentĴ scales asĴ ∝εμV 2 /L 2λ s , whereV is the applied voltage,ε is the permittivity andμ is the electric mobility, in contrast to the familiar diffusionfree scalingĴ ∝εμV 2 /L 3 . Thus, diffusion is seen to lead to a large O(L/λ s ) increase in current. Finally, we derive an asymptotic recombination-voltage relation for a kinetically limited OLED, in which charge recombination occurs on a much longer time scale than diffusion and drift.