He enrolled in the faculty of engineering in Milan in 1923, but after moving to Rome in 1925 he graduated with honors in architecture at Sapienza in 1928. In 1929 he won the competition organized by the Institute for Popular Housing in Rome to build a model house in Garbatella (lot XXIV). He approaches the demands of rationalism and in 1930 participates, together with Bottoni and Pollini, as an Italian delegate at the third CIAM congress in Brussels. While not adhering to MIAR, he collaborates in the organization and participates in the Rational Architecture Exhibitions of 1928, 1931 and 1932. In 1933 he participated in the V Triennale di Milano with the two projects for the high house with a steel structure, also known as the Genoese high house and for the shelter La stamberga dei 12 skiers. In the period 1931-1933 he collaborated with the detached section of the Superintendency of Turin and was Honorary Inspector for the Monuments of Liguria. In this role in 1932 he designed the New Andrea Doria Maritime Station in Genoa;  he subsequently designed the Casa del Fascio in Rapallo (1937) and two other Case del Fascio also in Liguria. In 1934 he participated with Antonio Carminati, Pietro Lingeri, Ernesto Saliva, Giuseppe Terragni in the first degree competition for the Palazzo del Littorio in Rome and in 1935 in that for the new Auditorium in Rome, on the site now occupied by FAO (formerly the new headquarters of the Ministry of the Colonies), with a very bold project, not awarded by the jury. In 1934, together with his friend Ignazio Gardella, he also participated in the competition for the construction of the Casa del Fascio in Oleggio (Novara) in which he distinguished himself, although not winning, for the modernity of the Lecorbusierian school of the project. For the realization of the Universal Exposition of 1942, in 1936, the Commissioner Vittorio Cini presented Mussolini with a list of the greatest names in architecture of the time including names such as Libera, Del Debbio, Terragni, Michelucci, Montuori and Muzio; the Duce will choose Vietti together with Piacentini, Pagano, Piccinato and Rossi. In this project he approaches, helping to define, with the other architects of the E42, the rationalist current. His are the ultra-modern elevations of the glass buildings featured in the project presented to Mussolini, who, however, does not approve of this architecture because he considers it too modern and “devoid of figurative references to the history of national architecture".
The postwar period and “the image of the bourgeoisie"
In the postwar years he devoted himself to the construction of prestigious villas and homes in tourist sites, and in particular in Cortina d’Ampezzo, characterized by the search for social representativeness and a harmonious relationship with the surrounding nature. In Cortina he also deals with the town planning (1955-1956). In the same years he also worked on the town planning of Genoa (1959), of Sanremo and in the urban planning of the town of Portofino (1957-1960). Between 1950 and 1953 he was involved in the recovery project of the Island of San Giorgio Maggiore in Venice, on behalf of Count Vittorio Cini, who here created the Foundation dedicated to his son Giorgio, who died prematurely. On the island, Vietti contributes to the restoration of some existing buildings and carries out new projects, the best known of which is the Teatro Verde, which he signs together with Angelo Scattolin. In the same period he designed, again with Scattolin and Cesare Pea, the new headquarters of SADE, of which Cini is a shareholder, on the banks of the Rio Novo. After the death of Cini’s wife, Lyda Borelli, she will design the Nido Verde in Rome, in the Monte Mario area, named after her and inaugurated in 1961. He is mentioned in Michelangelo Antonioni’s film The Night as the designer of the villa where the night party takes place; this is the Barlassina Golf Club, built since 1958 with John S. M. Morrison. He receives the assignment for the offices of the Banca Popolare di Novara and precisely in Savona, Genoa, Sanremo, Vigevano, Brescia, Milan, Naples, Verona (1958-1979).
In 1964 he created in Massawa, Eritrea, the Villa Melotti on the Red Sea called the Cypraea, then destroyed in 2006 by the Eritrean dictatorial government.
The Costa Smeralda: The village of Porto Cervo in Gallura.
At the beginning of the sixties, Prince Karim Aga Khan IV commissioned him, together with Jacques Couelle and Michele Busiri Vici, to design his interventions on the Costa Smeralda.  In this context, he created the center of the village of Porto Cervo, hotels, such as the Cervo and the Pitrizza, the complex of the Dolce Sposa and many prestigious villas such as the Cerbiatte, chosen by Prince Karim to make it his home, and the Romazzine on the promontory. homonym. Its also the complexes of Cala Granu and Cala del Faro.
Still within the Costa Smeralda project he participated with his inspiration in the creation of Cerasarda ceramics with decorations that have characterized many of the most typical furnishings of the houses and hotels of the Costa. He was also, with his old ship Tamory, one of the founders of the Yacht Club Costa Smeralda. The great esteem and friendship with the publishers Gjlla Giani and Giuseppe Maria Jonghi Lavarini, founders of the Di Baio Editore publishing house, meant that he published with them the most substantial anthology of his projects. The publishing house has many manuscripts and images of the architect. He receives the honorary citizenship of the city of Cannobio in 1996. He dies in Milan on 28 March 1998, at the age of 95.