Darius Milhaud was the member of Les Six who composed the most works for cello. The Violoncello Concerto No. 1, Op. 136 is not only his earliest cello work, but also the most popular. There are eight chapters in this document. The first four chapters include an overview of Darius Milhauds life, musical style, and achievement, major cello works, and a discussion about the relationship between him and cellist Maurice Marchal, who was the dedicatee of this concerto. The main body of this document is an analysis of this cello concerto. This analysis demonstrates Milhauds use of polytonal concept by using polymodality and polychords, and reveals Milhauds musical ambiguity regarding harmonic, melodic, and rhythmic materials, as well as formal structure. This analysis also includes a comparison of the orchestral score and the piano reduction to provide a reference for the cellist performer. In the analysis of each movement, comparison is made between this cello concerto and Milhauds Violoncello Concerto No. 2, Op. 255.