The radiocarbon dating method has been claimed to provide considerable support for evolutionary theories of the past which conflict with the Biblical record of the Earth's early history. This paper seeks to answer the question: how can the radiocarbon activity measurements be understood in a way that is consistent with the Biblical framework of history? The evidences for the non-equilibrium theory of radiocarbon variations are discussed and are shown to give the theory a sound theoretical foundation. The prediction that atmospheric C-14 activities have increased with time has been tested by studying the results obtained from known-age samples. Consistency between theory and observation is found for the period of the last 2600 years, but not before this. It follows that either the non-equilibrium model is wrong, or that the chronologies of Ancient Egypt and of the tree-ring sequences are in need of major revision. Evidences from Biblical archaeology which strongly suggest the need of a revision are briefly discussed. Using a revised chronological scheme, the C-14 activities of the archaeological samples have been recalculated, and it is found that the results are consistent wit the non-equilibrium prediction. Consequently, the theory is self-consistent, and this promotes confidence in the general approach. The form of a creationistic calibration curve for C-14 dates is suggested, so that use may be made of the dating system when re-evaluating the facts relevant to prehistory.