In a similar fashion to diffusers or other highly scattering media, multimode fibres deliver coherent light signals in the form of apparently random speckled patterns. In contrast to other optically random environments, multimode fibres feature remarkably faithful cylindrical symmetry. Our experimental studies challenge the commonly held notion that classifies multimode fibres as unpredictable optical systems. Instead, we demonstrate that commercially available multimode fibres are capable of performing as extremely precise optical components. We show that, with a sufficiently accurate theoretical model, light propagation within straight or even significantly deformed segments of multimode fibres may be predicted up to distances in excess of hundreds of millimetres. Harnessing this newly discovered predictability in imaging, we demonstrate the unparalleled power of multimode fibre-based endoscopes, which offer exceptional performance both in terms of resolution and instrument footprint. These results thus pave the way for numerous exciting applications, including high-quality imaging deep inside motile organisms.