The authors present an algorithm to compute the Wedderburn decomposition of semisimple group algebras FG based on the computational approach of the Brauer-Witt theorem. The algorithm has been implemented in the GAP package Wedderga. The paper includes all the theoretical background. For this we refer also to [G. Olteanu, Wedderburn decomposition of group algebras and applications. Cluj-Napoca: Editura Fundaţiei Pentru Studii Europene (2008; Zbl 1165.16001) and Math. Comput. 76, No. 258, 1073-1087 (2007; Zbl 1117.20004)] and to [A. Olivieri, Á. del Río, J. J. Simón, Commun. Algebra 32, No. 4, 1531-1550 (2004; Zbl 1081.20001)] and [A. Olivieri, Á. del Río, J. Symb. Comput. 35, No. 6, 673-687 (2003; Zbl 1050.16015)]