The curling of a graphitic sheet to form carbon nanotubes produces a class of materials that seem to have extraordinary electrical and mechanical properties. In particular, the high elastic modulus of the graphite sheets means that the nanotubes might be stiffer and stronger than any other known material, with beneficial consequences for their application in composite bulk materials and as individual elements of nanometre-scale devices and sensors. The mechanical properties are predicted to be sensitive to details of their structure and to the presence of defects, which means that measurements on individual nanotubes are essential to establish these properties. Here we show that multiwalled carbon nanotubes can be bent repeatedly through large angles using the tip of an atomic force microscope, without undergoing catastrophic failure. We observe a range of responses to this high-strain deformation, which together suggest that nanotubes are remarkably flexible and resilient.