Mather was born in Baoding, China and grew up there until he came to United States to go to college, graduating summa cum laude in 1935 from Princeton University. His plans to return to China were interrupted by the war and he instead went on to the University of California, Berkeley, to pursue his PhD in Chinese literature, studying with Peter Boodberg and others.

Mather came to the University of Minnesota in 1949 to found the study of Chinese language and literature. In the following decades, he was a major force in Chinese studies at the university and across the nation. He was central to establishing the field of early medieval Chinese studies with his monumental translation, A New Account of Tales of the World. Even after his 1984 retirement Mather was very active, publishing The Poet Shen Yüeh: The Reticent Marquis and the two-volume The Age of Eternal Brilliance: Three Lyric Poets of the Yung-ming Era. His New Account was reissued in a revised second edition by U of Michigan Press in 2002.

His beloved wife Virginia, to whom the Age of Brilliance was simply dedicated as "For Ginny," passed away in 2012. Richard currently lives with his daughter in Northern Minnesota.