Fano studied at the University of Turin which he entered in 1888. His studies there were directed by Corrado Segre and he was also influenced by Castelnuovo. In fact Castelnuovo had been appointed as D'Ovidio's assistant in Turin the year before Fano began his studies and Corrado Segre had been appointed to the chair of higher geometry in Turin the year that Fano entered the University of Turin. This was an exciting place for research in geometry and it is not surprising that Fano was led to specialise in this area.
In 1892 Fano graduated from Turin and then, in 1893, he went to Göttingen to undertake research and of course to study under Felix Klein. Twenty years earlier, in 1872, Klein had produced his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872). The Erlanger Programm gave the unified approach to geometry that is now the standard accepted view. Corrado Segre corresponded regularly with Klein and in this way Fano had been brought to Klein's attention. In fact this had led to Fano translating the Erlanger Programm into Italian while he was an undergraduate at Turin and it had been published in the Annali di matematica in 1890. Of course Fano did indeed study with Klein in Göttingen where he attended his lectures.
Fano became Castelnuovo's assistant in Rome in 1894, a position he held for four years. Following this assistantship, Fano went to Messina in the extreme northeastern Sicily where he worked from 1899 to 1901. He had left that city well before an earthquake struck Messina on 28 December 1908, almost totally destroying the city and killing 78000. By this time Fano was far away in Turin where he had been appointed as professor at the University in 1901. In 1911 Fano married Rosetta Cassin and their two children, both of the sons, became professors in the United States.
He taught at Turin from 1901 until 1938 when he was deprived of his chair by the Fascist Regime. After this Fano went to Switzerland where he taught Italian students at an international camp near Lausanne. After the end of World War II Fano, who was seventy-four years old by this time, continued to travel and lecture on mathematics. In particular he visited the United States where he lectured and he also lectured in his native Italy during the remaining six years of his life.
Fano's work was mainly on projective and algebraic geometry. Fano was a pioneer in finite geometry and one of the first people to try to set geometry on an abstract footing. Before Hilbert was to make such abstract statements Fano said:-
As a basis for our study we assume an arbitrary collection of entities of an arbitrary nature, entities which, for brevity, we shall call points, but this is quite independent of their nature.Struik, in , describes Fano's contribution:-
Early studies deal with line geometry and linear differential equations with algebraic coefficients ... . Later work is on algebraic and especially cubic surfaces, as well as on manifolds with a continuous group of Cremona transformations. He showed the existence of irrational involutions in three-space S3, i.e., of "unirational" manifolds not birationally representable on S3. He also studied birational contact transformations and non-euclidean and non-archimedean geometries.Fano wrote many textbooks, examples of which are his famous geometry texts Lezioni di geometria descrittiva (1914) and Lezioni di geometria analitica e proiettiva (1930). This last text was written jointly with Alessandro Terracini who wrote the obituaries  and  of Fano.
Article by: J J O'Connor and E F Robertson