Underwater wireless communication (UWC) technology has attracted widespread attention in the past few years. Compared with conventional acoustic underwater wireless communication technology, underwater optical wireless communication (UOWC) technology has promising potential to provide high data rate wireless connections due to the large license-free bandwidth. Building a high-performance and reliable UOWC system has become the target of researchers and various advanced and innovative technologies have been proposed and investigated. Among them, better hardware such as transmitters and receivers, as well as more advanced modulation and signal processing techniques, are key factors in improving UOWC system performance. In this paper, we review the recent development in UOWC systems. In particular, we provide a brief introduction to different types of UOWC systems based on channel configuration, and we focus on various recent studies on advanced signal processing methods in UOWC systems, including both traditional non-machine learning (NML) equalizers and machine learning (ML) schemes based on neural networks. In addition, we also discuss the key challenges in UOWC systems for future applications.