Yang was born in Hefei, Anhui, China; his father, Yang Wu-Chih (楊克纯; courtesy name: 武之) (1896–1973), was a Mathematician, and his mother, Luo Meng-hua (羅孟華), was a housewife. Yang attended elementary school and high school in Beijing, and in the autumn of 1937 his family moved to Hefei after the Japanese invaded China. In 1938 they moved to Kunming, Yunnan, where the National Southwestern Associated University, Lianda, was located. In the same year, as a second year student, Yang passed the entrance examination and studied at Lianda. He received his bachelor's degree in 1942, with his thesis on the application of group theory to molecular spectra, under the supervision of Ta-You Wu. He continued to study graduate courses there for two years under the supervision of Wang Zhuxi, working on statistical mechanics. In 1944 he received his master's degree from Tsinghua University, which had moved to Kunming during the Sino-Japanese War (1937–1945). Yang was then awarded a scholarship from the Boxer Indemnity Scholarship Program, set up by the United States government using part of the money China had been forced to pay following the Boxer Rebellion. His departure for the United States was delayed for one year, during which time he taught in a middle school as a Teacher and studied field theory.
From 1946, Yang studied with Edward Teller (1908–2003) at the University of Chicago, where he received his doctorate in 1948. He remained at the University of Chicago for a year as an assistant to Enrico Fermi. In 1949 he was invited to do his research at the Institute for Advanced Study in Princeton, New Jersey, where he began a period of fruitful collaboration with Tsung-Dao Lee. He was made a permanent member of the Institute in 1952, and full professor in 1955. In 1963, Princeton University Press published his textbook, Elementary Particles. In 1965 he moved to Stony Brook University, where he was named the Albert Einstein Professor of Physics and the first Director of the newly founded Institute for Theoretical Physics. Today this institute is known as the C. N. Yang Institute for Theoretical Physics.
Yang married Chih-li Tu (Chinese: 杜致禮; pinyin: Dù Zhìlǐ), a Teacher, in 1950 and has two sons and a daughter with her: Franklin Jr., Gilbert and Eulee. His father-in-law was a Kuomintang Army General Du Yuming who was taken POW at the end of Chinese civil war. First wife Tu died in the winter of 2003. Yang married then 28-year-old Weng Fan (Chinese: 翁帆; pinyin: Wēng Fān) in December 2004.
He has been elected a Fellow of the American Physical Society, the Chinese Academy of Sciences, the Academia Sinica, the Russian Academy of Sciences, and the Royal Society. He was awarded honorary doctorate degrees by Princeton University (1958), Moscow State University (1992), and the Chinese University of Hong Kong (1997).
Yang visited the Chinese mainland in 1971 for the first time after the thaw in China–US relations, and has subsequently made great efforts to help the Chinese physics community rebuild the research atmosphere which was destroyed by the radical political movements during the Cultural Revolution. After retiring from Stony Brook he returned as an honorary Director of Tsinghua University, Beijing, where he is the Huang Jibei-Lu Kaiqun Professor at the Center for Advanced Study (CASTU). He also is one of the two Shaw Prize Founding Members and is a Distinguished Professor-at-Large at the Chinese University of Hong Kong.
He retired from Stony Brook University in 1999, assuming the title Emeritus Professor. In 2010, Stony Brook University honored Yang's contributions to the university by naming its newest dormitory building C. N. Yang Hall.
Yang became a U.S. citizen in 1964. He now resides in China, and he was granted permanent residency in China in 2004. He renounced his U.S. citizenship as of Sep 30, 2015 and became a citizen of China.