Soulima grew up in Paris, France, where the family had settled. There he studied composition and theory with renowned composer, conductor, and teacher, Nadia Boulanger, who had been friends with Igor Stravinsky for years. Soulima also studied piano with Isidor Philipp, French teacher, pianist, composer, and editor, at the National Conservatory of Music in Paris.
Soulima gave his first concert in 1934 in Paris. He became an important interpreter of his father's works. He concertized all over the world, performing mostly Igor's compositions, and some of his own. Father and son also performed and toured together. In 1931, Igor wanted to write a duo piano piece that he could perform with his son. The piece, the Concerto for two solo pianos, was finished on November 9, 1935, and father and son gave the premiere performance in Paris, on November 21, 1935. The concerto differs from the usual arrangement of a concerto, as there is no orchestra. Said Igor, "I wished to incorporate the orchestra and do away with it. The concerto was intended as a vehicle for concert tours in orchestra-less cities."
Igor Stravinsky moved to the United States in 1939, but Soulima stayed behind to join the French army. He lived in Europe until 1948, then moved to the U.S. and lived in both New York and Los Angeles. Soulima again began to play with his father, and performed the Capriccio for piano and orchestra, with Igor conducting. Igor later reduced the full orchestral score for two pianos (the original manuscript is in the Nadia Boulanger collection). Soulima continued to tour, and stretched his repertoire to play pieces other than his father's, such as those of Scarlatti and Mozart.
In 1950, Soulima Stravinsky became a faculty member at the University of Illinois School of Music where he taught piano until 1978. He performed faculty recitals that were recorded and released on the Centaur label. Some of his own compositions, such as his Sonata for cello and piano, and his three string quartets, were also released, along with Soulima Stravinsky Plays Igor Stravinsky. This recording consists of Soulima playing piano pieces, and some orchestral reductions of scores, such as, Three Movements from Petrushka, and Two Excerpts from L'Histoire du Soldat.
Soulima continued to concertize during the summer months, and also went back to visit Nadia Boulanger at the American Conservatory at Fontainebleau to play his father's pieces. In 1975, Soulima gave a piano concert in the Salle des Colonnes at Fontainebleau and played an all Igor Stravinsky program consisting of Serenade en La; Quatre Etudes, Op. 7; Sonate; Suite de l'Oiseau de Feu (with a transcription by Soulima); Les Cinq Doigts; Piano Rag-Music; Tango; and Trois mouvements de Petrouchka. In concert, Soulima was technically a master pianist; however, he had a hard touch, a strict sense of rhythm, and although extremely talented, seemed to lack the brilliant musicality of his father. For years, Soulima was violinist Roman Totenberg's accompanist, and just before his death, Soulima was working with Totenberg on composing a collection of Mozart violin cadenzas.