This thesis reports on the use of scanning tunnelling microscopy to elucidate the atomic-scale electronic structure of a charge density wave, revealing that it has a d-symmetry form factor, hitherto unobserved in nature. It then details the development of an entirely new class of scanned probe: the scanning Josephson tunnelling microscope. This scans the Josephson junction formed between a cuprate superconducting microscope tip and the surface of a cuprate sample, thereby imaging the superfluid density of the sample with nanometer resolution. This novel method is used to establish the...