Studied in Torino with C. Segre, E. D'Ovidio, and in Göttingen with F. Klein . He graduated in Torino (under Segre's supervision) in 1892. Taught in Messina (since 1899) and Torino (since 1900), in Switzerland and in U.S.A. during the period of racial laws, then again in Torino since 1945. He was member of the Accademia Nazionale dei Lincei.

Fano his one of the main Italian scholars in the study of the classification of varieties of higher dimension. His work on foundations of geometry has been also important and he gave the first examples of finite planes ("piani di Fano").

Students: A. Terracini, B. Segre, E. Togliatti.

Main commemorations and studies: A. Terracini, Rend. della R. Acc. Nazionale dei Lincei , (8), 14, 1953, 702-715; B. Segre, Archimede , 4, 1952, 262-263.