Endoscopy is a key technology for minimally-invasive optical access to deep tissues in humans and living animals. However, modern endoscopes, such as fiber bundles, still suffer from low spatial resolution. Multimode fiber is a very promising tool for high-resolution endo-microscopy. We use advanced wavefront shaping technique and experimentally demonstrate high-resolution fluorescent and label-free imaging through a multimode fiber. We also present an ultra-thin Raman imaging probe with an excellent ratio between field of view and probe diameter. However, state-of-the-art multimode fiber endo-microscopy still has several problems limiting its broad applications: slow speed, as well as requirements of complex wavefront shaping procedure and expensive spatial light modulators. Here we show the solution to all these problems. We propose and experimentally demonstrate a new method of high-resolution endoscopy: compressive multimode fiber imaging. The key idea is to integrate the compressive sensing technique with a multimode fiber probe, which produces a random basis of speckle patterns, collects the optical response and provides optical sectioning. This new approach allows high-speed diffraction-limited imaging at the full field of view of a probe and does not require complex elements, such as spatial light modulators or knowledge of the transfer matrix of the multimode fiber. We demonstrate high-resolution imaging through a fiber probe with the total number of measurements 20 times less than required for the standard raster scanning approach. Compressive multimode fiber imaging offers a unique tool for in vivo high-speed high-resolution endoscopy.