After several months of rehearsals, the quartet had its debut in Szeged in 1919. Soon the Budapest critics also took notice of their dynamic and mature playing. A year later Maurice Ravel was among those present at their Vienna debut, and was so captivated by the young Hungarian musicians that he immediately invited them to Paris. That launched their series of successes of the twenties and thirties. In 1923 the quartet settled in England, and it was from here that they went on their European, and later overseas, tours. They also made numerous recordings in London, some of which – among them the cycle of Beethoven quartets – won prestigious awards and sold hundreds of thousands of copies. "Our gramophone records are kept in the Beethoven Museum in Bonn and in the library of the White House in Washington, and we keep touring worldwide" – Léner told the Budapest Pesti Napló in November, 1931. "Our affairs are handled by eight impresarios and three secretaries; we have ninety-six engagements already booked for this season, and this number will increase even further." The recognition given to the ensemble is shown by the fact that Respighí, Milhaud, and Malipiero all dedicated one of their quartets to the Léner Quartet, and that Léner's methodological work, The Technique of String Quartet Playing, published in London, was taught for decades at important universities.
At the end of the thirties the quartet, in order to escape the war in Europe, moved to Mexico City, and then, in 1941, broke up. Léner, however, a year later reconstituted the quartet, by then based in New York, with Laszló Steinhardt playing second violin, Ralph Hirsch viola and Gábor Rejtő cello. The ensemble, which was successful primarily in America, at various times had as its members Miklós Harsányi, Ottó Déri, Mihály Kuttner, László Varga and Alfréd Indig. In 1947 and 1948 the quartet, comprising Léner, Kuttner, Harsányi and Varga, gave highly successful concerts in Budapest, but by then Léner was seriously ill and on 29 November 1948 died in New York.