This article concerns receiver design for light-emitting diode (LED) communications, where the LED nonlinearity can severely degrade the performance of the system. We propose extreme learning machine (ELM)-based noniterative and iterative receivers to effectively handle the LED nonlinearity and memory effects. For the iterative receiver design, we also develop a data-aided receiver, where data are used as virtual training sequence in ELM training. It is shown that the ELM-based receivers significantly outperform conventional polynomial-based receivers. Iterative receivers can achieve huge performance gain compared to noniterative receivers, and the data-aided receiver can reduce training overhead considerably. This article can also be extended to radio frequency communications, e.g., to deal with the nonlinearity of power amplifiers.