Trautman was born in Warsaw, Poland into an artistic family. His father, Mieczysław, was a painter and taught drawing at a secondary school in Warsaw. His mother, Eliza Trautman (née André), was French, though she was born in Spain, where her father, Marius André, was working as a French consular officer. His schooling, at the elementary level, was interrupted by the Warsaw Uprising of 1944. After spending about ten months in Germany, he returned, with his mother (his father had died in 1941) to Poland. In the fall of 1945, they both went to Paris, to stay with their family there. In France, Trautman attended a Polish secondary school from which he graduated in 1949 and returned to Poland shortly afterwards.
During the years 1949-55, he studied radio engineering at the Warsaw University of Technology. After earning a Master's degree under the influence of Warsaw Tech's theoretical physics professor Jerzy Plebański, he continued graduate work in Leopold Infeld's group at the Institute of Theoretical Physics of Warsaw University. This Institute was to become his permanent place of study and work.
Plebański suggested to Trautman to work on the problem of gravitational radiation. On the basis of that work, supervised also by Leopold Infeld, he obtained in 1959 the Ph.D. degree at the Institute of Physics of the Polish Academy of Sciences in Warsaw. Other early influences included Felix Pirani, Hermann Bondi, Ezra Ted Newman, Arthur Komar, Engelbert Schücking, and Roger Penrose.
Later, he was influenced by scientific contacts with André Lichnerowicz, Subrahmanyan Chandrasekhar, Chen Ning Yang and Abdus Salam. In 1960, Trautman and Ivor Robinson discovered an important family of exact solutions of the Einstein field equation, the Robinson-Trautman gravitational waves. Later, Trautman worked on the Einstein-Cartan theory of gravitation, Yang-Mills fields, and spinors on manifolds.
Trautman's doctoral students include the physicists M. Demiański, W. Kopczyński, J. Tafel, J. Sławianowski, J. Lewandowski, P. Nurowski and the well-known astrophysicists J. P. Lasota (now in Paris), and M. Abramowicz (now at Göteborg). Trautman is a member of the Polish Academy of Sciences (PAN) and of the Polish Academy of Arts and Sciences (PAU).