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Born Abram Solman Borowitz in New York City, Burrows graduated from New Utrecht High School in Brooklyn and later attended both City College and New York University. He began working as a runner on Wall Street while at NYU, and he also worked in an accounting firm. After he met Frank Galen in 1938, the two wrote and sold jokes to an impressionist who appeared on the Rudy Vallée radio program.

His radio career gained strength when he collaborated with Ed Gardner, the writer and star of radio legend Duffy's Tavern. The two created the successful series after Gardner's character, Archie, premiered on the earlier radio program, This Is New York. Burrows was made the show's head writer in 1941, and he credited the experience with investing the Runyonesque street characters he fashioned for Guys and Dolls. "The people on that show," Burrows once said about Duffy's Tavern, "were New York mugs, nice mugs, sweet mugs, and like (Damon) Runyon's mugs they all talked like ladies and gentlemen. That's how we treated the characters in Guys and Dolls."