Raman spectroscopy is an optical technique based on inelastic scattering of light by vibrating molecules and can provide chemical fingerprints of cells, tissues or biofluids. The high chemical specificity, minimal or lack of sample preparation and the ability to use advanced optical technologies in the visible or near-infrared spectral range (lasers, microscopes, fibre-optics) have recently led to an increase in medical diagnostic applications of Raman spectroscopy. The key hypothesis underpinning this field is that molecular changes in cells, tissues or biofluids, that are either the cause or the effect of diseases, can be detected and quantified by Raman spectroscopy. Furthermore, multivariate calibration and classification models based on Raman spectra can be developed on large "training" datasets and used subsequently on samples from new patients to obtain quantitative and objective diagnosis. Historically, spontaneous Raman spectroscopy has been known as a low signal technique requiring relatively long acquisition times. Nevertheless, new strategies have been developed recently to overcome these issues: non-linear optical effects and metallic nanoparticles can be used to enhance the Raman signals, optimised fibre-optic Raman probes can be used for real-time in-vivo single-point measurements, while multimodal integration with other optical techniques can guide the Raman measurements to increase the acquisition speed and spatial accuracy of diagnosis. These recent efforts have advanced Raman spectroscopy to the point where the diagnostic accuracy and speed are compatible with clinical use. This paper reviews the main Raman spectroscopy techniques used in medical diagnostics and provides an overview of various applications.