The current view of space-time chronometry is reviewed in its different elements, clock transport and the ‘twin paradox,’ the effect of a gravitational field, and the Doppler effect. In this view a class of privileged, local inertial frames is assumed, and all good clocks measure proper time, the only observable one needs. Relativistic effects on clock synchronization and frequency comparison are described, and the essential role of the geoid is stressed. The available experimental precision makes these effects an essential ingredient of current and future work in this field. The role of distant matter in determining inertia and local dynamics (‘Mach's principle’) is not adequately described in this view. The cosmic background radiation provides a precise definition and a measurement of the local standard of rest with respect to the universe, which might have a privileged character among all the inertial frames. Different kinds of violations of the conventional view based upon this conjecture are discussed in terms of free dimensionless parameters. Finally, we review the upper bounds to the discrepancies from the relativistic values as deduced from several experiments.