Among homologous series of metal oxides, Andersson-Magneli phases TinO2n-1 (n=4-10) have attracted renewed scientific attention because of their behaviour in electrical conductivity and chemical/thermal stability. Various applications have also been reported for the phases with different values of n, or slightly reduced rutile (TiO2). The characteristic properties of these materials depend strongly on the compositional deviation from TiO2 and the way in which the structure accommodates the deviation. Thus, an urgent requirement is to overcome difficulties in characterizing such materials at atomic resolution. Here, we trace the discovery of the Andersson-Magneli phases, and report the application of recent developments in electron microscopy to reveal the relation, at the local level, between structural characteristics and electronic states, specifically for the materials TinO2n-1 with n=4-8. The electrical conductivity of Ti4O7 has been reported previously to show three clearly distinct states on decreasing temperature from 300 K. For this reason, we focus on Ti4O7 as a representative example of the TinO2n-1 phases and report structural characteristics at temperatures corresponding to each of the three different phases, focusing on the distribution of Ti3+ and Ti4+ cations from analysis of single-crystal XRD data. Electron diffraction experiments and electrical conductivity measurements are also reported.