In 2014, present-day scientists had the opportunity of marking the centennial of a discovery that triggered the development of a new field of research, which is now called Solid State Ionics.In their 1914 paper, Carl Tubandt and Erich Lorenz reported on the extraordinary properties of the alpha phase of silver iodide. Although α-AgI was a crystalline material, it resembled a molten salt with regard to the liquid-like value and weak temperature dependence of its ionic conductivity. With their transference measurements, Tubandt and Lorenz proved that the electric current in α-AgI was completely carried by the silver ions, while the iodide ions formed a rigid lattice. Up to the present day, α-AgI has been considered the fast ion conductor par excellence.In the mid-1930s, L.W. Strock was the first to use x-ray diffraction to investigate the crystal structure of α-AgI. The anion sublattice was found to be body centered cubic, but the arrangement of the silver ions remained a puzzling question. On the one hand, Strock could assign a large number of possible crystallographic sites to them. On the other hand, the state of the silver ions appeared to be rather ‘quasi-molten’ or ‘liquid-like’. This structural puzzle was resolved in 1977, when Cava, Reidinger and Wuensch used the results of a single-crystal neutron-diffraction experiment to construct contour plots for the probability density of the silver ions in α-AgI, which turned out to have broad maxima at the tetrahedral voids of the anion structure, with saddle points between them.A number of novel experimental approaches toward a better understanding of the ion dynamics in α-AgI were suggested by Wilhelm Jost in the 1960s and 1970s. These included high-accuracy specific heat measurements, measurements of the ionic conductivity in the microwave and far-infrared frequency regimes, and quasielastic neutron scattering. The results of the ensuing experiments, involving the present author, did not always provide immediate answers to the long-standing open questions, but rather created new puzzles instead. In this Chapter, an overview is given of the essential steps that were taken in experiment and modeling, eventually leading to the emergence of a self-consistent picture of the structure and dynamics of the mobile silver ions in α-AgI. Notably, that picture included both solid-like and liquid-like aspects. Strictly speaking, however, either category, ‘solid’ and ‘liquid’, had to be considered inappropriate for characterizing the actual state of the silver-ion sublattice.Recently, the transition to a more solid-like behavior of the mobile silver ions was observed in low-temperature α-AgI, which could be stabilized by confinement in glass, as first shown by Tatsumisago et al. Far below the regular phase transition temperature, 147 °C, measurements were performed of the frequency-dependent conductivity of α-AgI, yielding relevant information on the silver-ion dynamics. A conjecture put forward by Jost in 1937 could thus be corroborated by the present author and his coworkers. Below 147 °C, the ‘liquid-like’ activation energy for ionic transport was found to be replaced by a larger, more ‘solid-like’ value, although the anion structure and, therefore, the barriers for elementary displacements of the cations remained essentially unchanged. The underlying mechanism is sketched at the end of the Chapter.