Born in Sheffield, England, Shaw immigrated to the U.S .at the age of eighteen. He quickly achieved success as a businessman, and in 1840, retired from his business pursuits to spend nearly a decade traveling the world and seeing gardens in Asia, Russia and Europe. Returning to St. Louis and inspired by his travels, he began construction of a botanical garden on his country estate, Tower Grove. In 1859 he opened the Missouri Botanical Garden to the public, which included a research facility for scientists that is today a world-renowned scientific institution. In 1868 Shaw began the development of Tower Grove Park adjacent to Tower Grove, designed in the Gardenesque style. The 280-acre park includes thousands of plants and trees and numerous pavilions, conservatories, and decorative follies. Believing that the study and enjoyment of garden botany could be useful to all as a means to reform society, and provide the benefits of education, health, and happiness, Shaw donated both properties to the City of St. Louis for the public’s use.
Shaw founded the School of Botany at Washington University in St. Louis and was a founding member of the Missouri Historical Society. He authored two books, The Rose (1879), and The Vine and Civilization (1884).