Recent technological developments have resulted in the availability of miniaturised spectral imaging sensors capable of operating in the multi- (MSI) and hyperspectral imaging (HSI) regimes. Simultaneous advances in image-processing techniques and artificial intelligence (AI), especially in machine learning and deep learning, have made these data-rich modalities highly attractive as a means of extracting biological information non-destructively. Surgery in particular is poised to benefit from this, as spectrally-resolved tissue optical properties can offer enhanced contrast as well as diagnostic and guidance information during interventions. This is particularly relevant for procedures where inherent contrast is low under standard white light visualisation. This review summarises recent work in surgical spectral imaging (SSI) techniques, taken from Pubmed, Google Scholar and arXiv searches spanning the period 2013-2019. New hardware, optimised for use in both open and minimally-invasive surgery (MIS), is described, and recent commercial activity is summarised. Computational approaches to extract spectral information from conventional colour images are reviewed, as tip-mounted cameras become more commonplace in MIS. Model-based and machine learning methods of data analysis are discussed in addition to simulation, phantom and clinical validation experiments. A wide variety of surgical pilot studies are reported but it is apparent that further work is needed to quantify the clinical value of MSI/HSI. The current trend toward data-driven analysis emphasises the importance of widely-available, standardised spectral imaging datasets, which will aid understanding of variability across organs and patients, and drive clinical translation.