Professor Begić graduated from the Law Faculty in the University of Sarajevo in 1976 and earned his Master’s degree in law from the University of Belgrade in 1979. He earned his PhD from the same institution in 1981.  
In 1976, he was appointed as an assistant lecturer at the Law Faculty in the University of Sarajevo.  He was promoted to the position of senior lecturer in 1981, then to associate professor in 1987 and became full professor in 1995, teaching the subjects Finances and Financial Law, and Economic Policy.

Не was а member as well as the President of the Legislative Commission in the Assembly of Bosnia аnd Herzegovina (between 1990 and 1996), and also а member of the Constitutional Commission of the Constituent Assembly of the Federation of Bosnia аnd Herzegovina. Between 1993 and 1996, he was part of the delegation of Bosnia and Herzegovina in numerous реасе negotiations under the auspices of the international community, including the negotiations in Dayton, Ohio, which resulted in the General Framework Agreement for Реасе in Bosnia and Herzegovina. He has authored a number of textbooks, treatises and monographs.

Не served as a Judge of the Constitutional Court of Bosnia and Herzegovina from May 1997 to May 2002. He served as the President of the Constitutional Court from June 1999 to February 2001.