Underwater wireless communications refer to data transmission in unguided water environment through wireless carriers, i.e., radio-frequency (RF) wave, acoustic wave, and optical wave. In comparison to RF and acoustic counterparts, underwater optical wireless communication (UOWC) can provide a much higher transmission bandwidth and much higher data rate. Therefore, we focus, in this paper, on the UOWC that employs optical wave as the transmission carrier. In recent years, many potential applications of UOWC systems have been proposed for environmental monitoring, offshore exploration, disaster precaution, and military operations. However, UOWC systems also suffer from severe absorption and scattering introduced by underwater channels. In order to overcome these technical barriers, several new system design approaches, which are different from the conventional terrestrial free-space optical communication, have been explored in recent years. We provide a comprehensive and exhaustive survey of the state-of-the-art UOWC research in three aspects: 1) channel characterization; 2) modulation; and 3) coding techniques, together with the practical implementations of UOWC.