This is a lively memoir about growing up with the charismatic American astronomer and science impresario, Harlow Shapley, by his daughter, the late Mildred Shapley Matthews. Shapley remains widely regarded as one of the most unusual, interesting, and noteworthy American astronomers, internationalists, and humanitarians of the 20th century. The "round table" in the title refers to a large rotating wooden desk mounted on central spindle, which graced the Director's Office at the Harvard College Observatory from 1906 through the mid-1950s. Mildred Shapley Matthews (1915-2016) wrote this reminiscence of life with her father, Harlow Shapley (1885-1972), during the early 1960s. It is based on her personal recollections plus extensive correspondence and conversations with her father and her mother, Martha Betz Shapley (1890-1981). Written in a colloquial narrative style, it exhibits for the first time a delightfully human side of Harlow Shapley. Harlow Shapley was also an outspoken political progressive, socialist, pacifist, and internationalist who openly espoused his political and social causes. He was one of the "suspected communists" named by Senator Joseph McCarthy, and in 1946 was called before the House Un-American Activities Committee. Shapley retired as director of the Harvard Observatory in 1952. Over the next 20 years he wrote several books and lectured extensively on college campuses across the country, sharing his enthusiasm with generations of future thinkers about the wonders of science, and his witty disdain for humankind's hubris.