Optical wireless communications are realized by modulating the intensity of a light source and detecting intensity fluctuations at the receiver. This mode of operation, known as intensity modulation and direct detection (IM/DD), is simple to implement in practice. However, computing the channel capacity of the underlying channel is not straightforward because of the amplitude constraints that arise due to IM/DD operation. In particular, the transmit signal must be non-negative, while the peak and average amplitudes are constrained due to practical and safety considerations. Though a closed form for the capacity of IM/DD channels is not known, much work has been done to find capacity bounds and asymptotic capacity expressions. In this paper, a description of the IM/DD channel and its physical constraints is presented, followed by a review of recent progress pertaining to the capacity of IM/DD channels. Additionally, capacity-achieving distributions are discussed along with simple constructions that approach capacity. This article is part of the theme issue ‘Optical wireless communication’.