This is among the first works Hindemith completed after immigrating to the United States in 1940. He had been teaching in Turkey for some years, and occasionally visited the U.S. Upon arriving to stay he was fortunate enough to go first to Tanglewood, where he would have a summer position teaching the cream of the crop of young American musicians. In the week before the session opened, he wrote his wife, had ample opportunity to see the area's beautiful mountains and forests, to make considerable progress on his new cello concerto, and in general develop positive feelings towards his future life in America.

He discovered that Serge Koussevitzky, director of Tanglewood and of the Boston Symphony Orchestra, had been championing his music, and showed him the completed parts of the score. Koussevitzky called the cellist Gregor Piatigorsky, who had happened to come to visit at Tanglewood that day, and showed it, and the two immediately scheduled it for performances in Boston, New York, and elsewhere on tour.

Hindemith had almost finished the new concerto a few weeks later, then got a different idea for the finale, scrapped the old one, and composed another. The finished work was premiered as scheduled by Piatigorsky and Koussevitzky, and has taken is place as an important twentieth-century cello concerto.

The concerto is habitually referred to with reference to its year of composition to distinguish it from an earlier concerto written in 1916 as a student work and from the Kammermusik No. 3, which Hindemith also designated as a cello concerto.

The first movement begins with an arresting march figure and is forceful throughout, with assertive sections for cello, scored lightly and colorfully in an acutely well-judged understanding of the balances that are possible for the cello and orchestra. They are interleaved with tutti passages that must have been inspired by the Boston Symphony's outstanding brass section.

The second movement begins in contemplative mood, with a lyrical theme (later used by William Walton in his Variations on a Theme by Hindemith). Perhaps remembering the weak tarantella with which he had ended his student cello concerto, Hindemith then gives the cello a tarantella theme in a fast middle section. After it, the tempo slows again, the orchestra returns to the opening material, but the cello keeps going with the tarantella at the previous speed. The music thus proceeds at two levels that only reach a moment of agreement at the end of the movement.

The second draft of third movement, based on a theme supposedly by Princess Anna Amalia of Prussia, sees the cello also seeming to try to stand apart from the orchestra. The orchestra keeps breaking into a march, which the cello resists. At the end, however, the cello seems to agree to conform to the orchestra's tempo and rhythm, adopts a jaunty repeating phrase, and seems to march away.

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