The precession-nutation transformation describes the changing directions on the celestial sphere of the Earth's pole and an adopted origin of right ascension. The coordinate system for the celestial sphere is the geocentric celestial reference system, and the two directions are the celestial intermediate pole (CIP) and the celestial intermediate origin (CIO), the latter having supplanted the equinox for this purpose following IAU resolutions in 2000. The celestial coordinate triad based on the CIP and CIO is called the celestial intermediate reference system; the prediction of topocentric directions additionally requires the Earth rotation angle (ERA), the counterpart of Greenwich sidereal time (GST) in the former equinox based system. The purpose of this paper is to review the different ways of calculating the CIP and CIO directions to precisions of a few microarcseconds over a time span of several centuries, meeting the requirements of high-accuracy applications.