Robert Bishop was born in Philadelphia, and was educated in public schools in Swarthmore, Pennsylvania after his family moved there from the city. He stayed in Swarthmore for college, earning his B.A. with Phi Beta Kappa honors in 1930, and also lived there in part of his adult life. His architectural training began in the office of W. Pope Barney, and in evening classes at the Drexel Institute and with the T-Square Club. Inspired by lectures given by Frank Lloyd Wright at the University of Pennsylvania in 1932, Bishop went to Taliesen in Wisconsin, where he remained as a fellow for three years. His later employer and collaborator in the creation of the Bryn Gweled community in Bucks County, Paul Beidler, was at Taliesen concurrently. While there, Bishop participated in producing the Broadacre City model. Wright's approach to natural materials and idealistic community planning and housing design would remain important features of his work through his career. Bishop's architectural interests were based in and informed by the social concerns of his Quaker faith. Bishop returned to Philadelphia after accompanying the completed Broadacre model to New York for exhibition in 1935.

Between his return to Philadelphia and the close of World War II, Bishop worked in a number of firms in the region, including Paul Beidler, Thomas & Martin, W. R. Morton Keast, and Baader, Young & Schultze. Bishop was also employed by R.C.A in this period, and by engineers Moody & Hutchison. In 1945, he formed a partnership with John W. Wright, which continued until 1951. In 1952, Bishop formed a new firm with Newcomb T. Montgomery, which they considered the successor to the practice of Bishop & Wright. In 1962, Montgomery & Bishop were joined by C. Treat Arnold. Bishop was a director of the Philadelphia Chapter of the AIA in 1951-2. He also became part of the faculty of the University of Pennsylvania's School of Fine Arts in 1952, where he was brought on as a lecturer by dean G. Holmes Perkins as part of the young and energetic teaching staff Perkins used to supplant the school's Beaux-Arts traditions with modernist progressivism. Bishop's first-hand knowledge of Wright's theories and his talent for designing with wood in Wright's vein were important contributions to the architectural curriculum in this formative period. Bishop continued to teach at Penn until 1960, where he had risen in rank from lecturer to Assistant Professor, and also served as a visiting critic at Cornell.

Bishop was one of the founders of Bryn Gweled (begun 1939), a cooperative homestead community in Bucks County, Pennsylvania, where he designed houses (along with Paul Beidler, Walter T. Robinson, Cornelius Bogert, and Thomas S. Michener) and made his residence. He retired from practice in 1972.

Written by Emily T. Cooperman.