Visible light communications (VLCs) have attracted considerable interest in recent years owing to the potential to simultaneously achieve data transmission and illumination using low-cost light-emitting diodes (LEDs). However, the high-speed capability of such links is typically limited by the low bandwidth of LEDs. As a result, spectrally efficient advanced modulation formats have been considered for use in VLC links in order to mitigate this issue and enable higher data rates. Carrierless amplitude and phase (CAP) modulation is one such spectrally efficient scheme that has attracted significant interest in recent years owing to its good potential and practical implementation. In this paper, we introduce the basic features of CAP modulation and review its use in the context of indoor VLC systems. We describe some of its attributes and inherent limitations, present related advances aiming to improve its performance and potential and report on recent experimental demonstrations of LED-based VLC links employing CAP modulation.