Astronomers have been consciously making positional observations of comets since the sixteenth century, and rough information on bright comets is available from social records collected in the Far East, the Middle East and Europe over the previous two millennia. Attempts at representing the observations with orbits also date back to the sixteenth century, well before the earliest computations were made on the basis of gravitational theory. Improvement in the accuracy of cometary astrometric data was slow to develop following the invention of the telescope and, a few decades later, the micrometer, for this had to await the availability of Flamsteed’s star catalogue and the understanding brought about by Bradley’s discovery of aberration and nutation. In spite of important contributions by some of the most celebrated scientists of the eighteenth century, orbit computation remained largely a process of trial and error until the discovery of the first asteroids and the rise of Gauss and his followers. By the 1880s the availability of a dense star catalogue like the AGK1 was allowing micrometric observations of comets and asteroids often to achieve an accuracy of 2 to 3 arcsec, and the subsequent introduction of photographic astrometry brought convenience but relatively little additional improvement in accuracy. The use nowadays of charge-coupled devices clearly has the potential for a further increase in accuracy, even for diffuse objects like comets, but for the moment the accuracy tends generally to be limited by the inadequacy of the available star catalogues. Modern orbit determination routinely includes the use of sophisticated techniques to identify isolated observations of the same asteroid at different oppositions and in the case of a comet the examination of the nongravitational effects that can strongly influence the object’s motion.