Although crystals are usually quite stable, they are sensitive to a disordered environment: even an infinitesimal amount of impurities can lead to the destruction of crystalline order. The resulting state of matter has been a long-standing puzzle. Until recently it was believed to be an amorphous state in which the crystal would break into 'crystallites'. But a different theory predicts the existence of a novel phase of matter: the so-called Bragg glass, which is a glass and yet nearly as ordered as a perfect crystal. The 'lattice' of vortices that contain magnetic flux in type II superconductors provide a good system to investigate these ideas. Here we show that neutron-diffraction data of the vortex lattice provides unambiguous evidence for a weak, power-law decay of the crystalline order characteristic of a Bragg glass. The theory also predicts accurately the electrical transport properties of superconductors; it naturally explains the observed phase transitions and the dramatic jumps in the critical current associated with the melting of the Bragg glass. Moreover, the model explains experiments as diverse as X-ray scattering in disordered liquid crystals and the conductivity of electronic crystals.