The 1914 Nobel Prize for Chemistry was awarded to Theodore Richards, whose work provided an insight into the history of the birth and evolution of matter as embedded in the atomic weights. However, the secret to unlocking the hieroglyphics contained in the atomic weights is revealed by a study of the relative abundances of the isotopes. A consistent set of internationally accepted atomic weights has been a goal of the scientific community for over a century. Atomic weights were originally determined by chemical stoichiometry--the so-called "Harvard Method," but this methodology has now been superseded by the "physical method," in which the isotopic composition and atomic masses of the isotopes comprising an element are used to calculate the atomic weight with far greater accuracy than before. The role of mass spectrometry in atomic weight determinations was initiated by the discovery of isotopes by Thomson, and established by the pioneering work of Aston, Dempster, and Nier using sophisticated mass spectrographs. The advent of the sector field mass spectrometer in 1947, revolutionized the application of mass spectrometry for both solids and gases to other fields of science including atomic weights. Subsequently, technological advances in mass spectrometry have enabled atomic masses to be determined with an accuracy better than one part in 10(7), whilst the absolute isotopic composition of many elements has been determined to produce accurate values of their atomic weights. Conversely, those same technological developments have revealed significant variations in the isotope abundances of many elements caused by a variety of physiochemical mechanisms in natural materials. Although these variations were initially seen as an impediment to the accuracy with which atomic weights could be determined, it was quickly realized that nature had provided a new tool to investigate physiochemical and biogeochemical mechanisms in nature, which could be exploited by precise and accurate isotopic measurements. Atomic weights can no longer be regarded as constants of nature, except for the monoisotopic elements whose atomic weights are determined solely by the relative atomic mass of that nuclide. Stable isotope geochemists developed mass spectrometric protocols by the adoption of internationally accepted reference materials for the light elements, to which measurements from various laboratories could be compared. Subsequently, a number of heavy elements such as iron, molybdenum and cadmium have been shown to exhibit isotope fractionation. The magnitude of such isotope fractionation in nature is less than for the light elements, but technological developments, such as multiple collector-inductively coupled plasma-mass spectrometry, have enabled such fractionation effects to be determined. Measurements of the atomic weights of certain elements affect the determination of important fundamental constants such as the Avogadro Constant, the Faraday Constant and the Universal Gas Constant. Heroic efforts have been made to refine the accuracy of the atomic weight of silicon, with the objective of replacing the SI standard of mass--the kilogram--with the Avogadro Constant. Improvements in these fundamental constants in turn affect the set of self-consistent values of other basic constants through a least-squares adjustment methodology. Absolute isotope abundances also enable the Solar System abundances of the s-, r-, and p-process of nucleosynthesis to be accurately determined, thus placing constraints on theories of heavy element nucleosynthesis. Future developments in the science of atomic weight determinations are also examined.