The results of the timing experiments using atomic clocks carried onboard circumnavigating airplanes [J. C. Hafele and R. E. Keating, Science 177, 168-172 (1971)] are reviewed. It is pointed out that the finding that the eastward-flying clock arrived back at the airport of origin with less elapsed time than its westward-flying counterpart was not expected based on the conventional interpretation of the Lorentz transformation (LT). The latter predicts that a moving clock always runs slower than one that is stationary in the rest frame of the observer. In order to satisfactorily represent the observed timing results it was necessary to assume that clocks slow in inverse proportion to γ=(1-vECM2c-2)-0.5≈0.5vECM2c-2, where vECM is the speed of the clock relative to the earthʼs center of mass (ECM) and c is the speed of light in free space. A similar result was obtained earlier in experiments in which an x-ray source and absorber were mounted on a high-speed rotor and the reference frame for computing clock speeds is the rotor axis. The common proportionality relationship for the two fundamental experiments is therefore referred to as the Universal Time-dilation Law (UTDL). Calculations of elapsed times based on the HK interpretation are presented which speak against the traditional LT interpretation of symmetric time dilation when clocks are not subject to unbalanced forces. A different space-time transformation is presented which is consistent with the UTDL, unlike the LT, but also satisfies both of Einstein's postulates of relativity and is compatible with the relativistic velocity transformation (RVT).