Abstract. During the late 1940s Harry Wiener introduced his “path number” as a descriptor of molecular structures, and demonstrated that it correlates highly with physico-chemical properties of alkanes. 25 years later Hosoya reformulated it in terms of graph theory. Since then, the Wiener index became highly popular both for diverse chemical applications, and as a basis for developing new graph-invariants. The presentation reviews these developments with the most important applications such as in the theory of molecular branching and cyclicity, in defining the properties of polymers and in detailed reconstruction of crystal growth. Theorems related to the Wiener number are presented. A variety of novel graph descriptors created in mathematical chemistry during the last 35 years are briefly examined, including distance polynomial, the Harary indices, the overall Wiener index, the Hyper-Wiener index, the Detour index, the Szeged and Cluj indices and their hyper-versions.