In order to deal with the increasing number of mobile devices and with their demand for Internet services, particularly social media platforms, streaming video, and online gaming, Radio-Frequency (RF) wireless networks have been pushed to their capacity limits. In addition to this, 80% of the total data traffic is carried out by users inside buildings. Therefore, new technologies have started to be considered for indoor wireless communications. Visible Light Communications (VLC) can provide both illumination and communications, appearing as an alternative or complement to RF wireless networks. VLC offers high bandwidth and immunity to interference from electromagnetic sources. This manuscript reviews recent high-capacity VLC demonstrations. The main focus of this work is to present digital-signal-processing techniques used in VLC systems. Different modulation formats are analyzed, which can be divided into two large groups, namely single-carrier and multi-carrier modulation schemes. Finally, some recently proposed capacity-achieving strategies are presented. We discuss how to implement these techniques and how they will be useful for the continued development of VLC systems.