Its foundation is largely due to the action of Francesco Severi, possibly starting from an idea by Luigi Fantappié.

The first Scientific Council was made up of Francesco Severi (president), Luigi Fantappiè, Giulio Krall, Enrico Bompiani and Mauro Picone.

From the beginning, the main activity of INdAM has been the organisation of advanced courses aimed at gifted young people.

The Italian mathematicians who worked as professors and/or were students at INdAM included Antonio Signorini, Gianfranco Cimmino, Iacopo Barsotti, Luigi Amerio, Beniamino Segre, Enzo Martinelli, Renato Caccioppoli, Fabio Conforto, Giovanni Battista Rizza, Aldo Andreotti, Edoardo Vesentini, Gaetano Fichera, Ennio De Giorgi, Claudio Procesi, Maurizio Cornalba, Alessandro Figà-Talamanca, Enrico Giusti, Antonio Ambrosetti, Paolo Marcellini, Enrico Bombieri, Corrado De Concini, Nicola Fusco and Mario Pulvirenti.

The foreign mathematicians included Leonard Roth, Helmut Hasse, Wilhelm Blaschke, Paul Dubreil, Lucien Godeaux, Luitzen Brouwer, Jean Leray, Wacław Sierpiński, Wolfgang Gröbner, Heinz Hopf, Erich Kähler, Oskar Zariski, Georges De Rham, Max Deuring, Bartel Leendert Van der Waerden, Kazimierz Kuratowski, John Lighton Synge, Louis Mordell, Rolf Nevanlinna, Richard von Mises, Ernst Witt, Henri Cartan, Jacques Tits, Jean Dieudonné, Victor Kac, Francis Clarke.