Born in Lindenwald, Kingdom of Prussia, Bauer developed a passion for art early on, despite his father's disapproval. After leaving home, he attended the Academy of Fine Arts and began working in abstract art in 1912. That same year, he met Herwarth Walden, founder of the magazine Der Sturm and its affiliated gallery, which became a key artistic center in Berlin. Bauer joined the Der Sturm group, which included artists like Wassily Kandinsky, Marc Chagall, Paul Klee, and Franz Marc, and he became a teacher at the Sturm School.

In 1916, Bauer met Baroness Hilla Rebay at Der Sturm. They became friends and lovers, though their relationship eventually turned platonic. Fourteen years later, Solomon Guggenheim and his wife, Irene, traveled to Germany with Rebay to meet Bauer and Kandinsky. By then, Bauer had shifted from lyrical to geometric abstraction. Guggenheim purchased several of Bauer's works and provided him with a stipend, allowing Bauer to open his own museum for non-objective art.

In 1937, Guggenheim established the Solomon R. Guggenheim Foundation, with Rebay as curator. Bauer traveled to the U.S. for the opening exhibition, which led to a solo show at the Arts Club of Chicago. Annual showings followed for the next four years.

Upon returning to Germany from an exhibition in Paris, Bauer was arrested by the Nazis for his "degenerate" art and selling works to Guggenheim. After several months in a Gestapo prison, he was released in August 1938 and emigrated to the U.S. in July 1939, just before World War II.

Bauer lived with Rebay before moving to Guggenheim's home in Deal, New Jersey. Guggenheim proposed a contract, which Bauer misunderstood. He signed, thinking he'd receive a lump sum for 110 paintings, but instead, he got a monthly stipend and had to leave future works to the foundation.

Feeling betrayed, Bauer stopped painting and had a falling out with Rebay, leading to a libel suit. In 1949, Solomon Guggenheim died, and the foundation's vision changed. Rebay was removed as curator, and Bauer's work was sent to storage.

Bauer died in 1953 of lung cancer in Deal, New Jersey. The Solomon R. Guggenheim Museum opened in 1959 without his works. His art remained unseen for two decades until a 1967 Guggenheim exhibition. A 1969 retrospective in Cologne and subsequent exhibitions revived interest in his work. In 2005, the Guggenheim mounted "Art of Tomorrow: Hilla Rebay and Solomon R. Guggenheim," featuring Bauer's work. A major retrospective took place at Weinstein Gallery, San Francisco in 2007.

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