© 2020 IEEE. Spectral efficient modulation formats can enable the transmission of higher data rates than conventional on-off keying (OOK). Carrierless amplitude and phase modulation (CAP) is such an attractive modulation scheme that has been widely considered for use in different types of optical links. The scheme however can suffer from intersymbol interference (ISI) and channel crosstalk (CCI) when the frequency response of the channel is not ideal. Conventional equalizers based on feedforward (FFE) and decision feedback (DFE) equalizers are easy to implement in practice and can mitigate some of the induced ISI. However, they fail to suppress the induced CCI in the link as each channel is equalized independently. As a result, we have recently proposed the use of a new equalizer structure for use in CAP-based optical links to mitigate these transmission impairments. This new equalizer, named CAP equalizer, can be formed with conventional FFEs and DFEs with minimal additional complexity whilst providing significant performance advantages. In this paper therefore, we review the equalizer structure and report recent demonstrations of its use in short-reach optical links. We present experimental studies on a 112 Gb/s CAP-16 VCSEL-based OM4 MMF link and a 4 Gb/s CAP-16 LED-based POF link and compare the performance of the links when both a conventional FFE and DFE equalizer and the newly proposed CAP equalizer are used. The results clearly demonstrate that the CAP equalizer offers improved receiver sensitivity and enables successful data transmission over longer fibre reaches.