Max Kaur studied law at Moscow International Independent University of Environmental and Political Sciences (IIUEPS), LL.B., and economics at Saint Petersburg State University of Engineering and Economics, MS in Economics. He earned his PhD in 2013 from Saint Petersburg State University of Economics.
Max Kaur was counsellor of the Mayor of Tallinn, Edgar Savisaar, from 2002 to 2005. He was vice-mayor of Maardu from 2005 to 2006. Max Kaur was first elected to Tallinn City Council in the Estonian local elections 2009. He was elected Mayor of Mustvee in April 2013. He was re-elected in December 2013.