Endomicroscopy is an emerging imaging modality, that facilitates the acquisition of in vivo, in situ optical biopsies, assisting diagnostic and potentially therapeutic interventions. While there is a diverse and constantly expanding range of commercial and experimental optical biopsy platforms available, fibre-bundle endomicroscopy is currently the most widely used platform and is approved for clinical use in a range of clinical indications. Miniaturised, flexible fibre-bundles, guided through the working channel of endoscopes, needles and catheters, enable high-resolution imaging across a variety of organ systems. Yet, the nature of image acquisition though a fibre-bundle gives rise to several inherent characteristics and limitations necessitating novel and effective image pre-and post-processing algorithms, ranging from image formation, enhancement and mosaicing to pathology detection and quantification. This paper introduces the underlying technology and most prevalent clinical applications of fibre-bundle endomicroscopy, and provides a comprehensive, up-to-date, review of relevant image reconstruction, analysis and understanding/inference methodologies. Furthermore, current limitations as well as future challenges and opportunities in fibre-bundle endomicroscopy computing are identified and discussed.