The scanning SQUID (Superconducting Quantum Interference Device) microscope is an extremely sensitive instrument for imaging local magnetic fields. We describe one such instrument which combines a novel pivoting lever mechanism for coarse-scale imaging with a piezoelectric tube scanner for fine-scale scans. The magnetic field sensor is an integrated miniature SQUID magnetometer. This instrument has a demonstrated magnetic field sensitivity of <10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−6</sup> gauss/√Hz at a spatial resolution of ∼10 µm. The design and operation of this scanning SQUID microscope are described, and several illustrations of the capabilities of this technique are presented. The absolute calibration of this instrument with an ideal point source, a single vortex trapped in a superconducting film, is shown. The use of this instrument for the first observation of half-integer flux quanta, in tricrystal thin-film rings of YBa <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> Cu <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> O <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7−δ</inf> , is described. The half-integer flux quantum effect is a general test of the symmetry of the superconducting order parameter. One such test rules out symmetry-independent mechanisms for the half-integer flux quantum effect, and proves that the order parameter in YBa <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> Cu <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> O <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7−δ</inf> has lobes and nodes consistent with d-wave symmetry.