Baxter was educated at Bancroft's School and Trinity College, Cambridge (BA, MA), before relocating to the Australian National University in Canberra to complete his PhD. He was among the first doctoral graduates in theoretical physics from the ANU, graduating in 1964. Then, in 1964 and 1965, he worked for the Iraq Petroleum Company. He worked as an assistant professor at the Massachusetts Institute of Technology from 1968 until 1970, when he took up a position at the ANU, and served a term as the Head of the Department of Theoretical Physics in the Institute of Advanced Study, until he retired in 2002. He is currently the Emeritus Professor of Physics. In 1984, he was awarded a Doctor of Science by Cambridge. He is a Fellow of the Australian Academy of Science, Royal Society of London, and the Isaac Newton Institute, Cambridge, where he was Royal Society Research Professor in 1992. In 1980 he was awarded the Boltzmann Medal, the main recognition for research contribution concerning statistical mechanics. In 2006, he was awarded the Lars Onsager Prize "For his original and groundbreaking contributions to the field of exactly solved models in statistical mechanics, which continue to inspire profound developments in statistical physics and related fields".
- Pawsey Medal, Australian Academy of Science, 1975
- Boltzmann Medal, IUPAP, 1980
- Thomas Ranken Lyle Medal, Australian Academy of Science, 1983
- Dannie Heineman Prize for Mathematical Physics, American Physical Society, 1987
- Harrie Massey Medal and Prize, Australian Institute of Physics / Institute of Physics (UK), 1994
- Fellow of the Australian Academy of Science, 1977
- Fellow of the Royal Society of London, 1982
- Centenary Medal, Australian Government, 2003
- Lars Onsager Prize, American Physical Society, 2006
- Royal Medal, 2013
- Henri Poincaré Prize, 2021
- V. Chari and A.N. Pressley, A Guide to Quantum Groups Cambridge University Press, 1994