The son of engineer Toivo Ilmari Leiviskä and teacher Sonja Jämsén-Astala, Leiviskä studied architecture at Helsinki University of Technology, qualifying as an architect in 1963. He established his own office in 1964, while also working as a teaching assistant at Helsinki University of Technology.
Leiviskä was made a member of the Royal Swedish Academy of Fine Arts in 1991. In 1992 he received Pro Finlandia Medal of the Order of the Lion of Finland, and was appointed as an 'Artist Professor' by the Finnish President. In 1994 he was made an Honorary Fellow of the American Institute of Architects and awarded the Prince Eugen Medal the same year. He was awarded the prestigious Carlsberg Prize in architecture in 1995. In 1997 Leiviskä followed Alvar Aalto and Reima Pietilä in becoming the architecture Member of the Academy of Finland - thus bestowing on him the title of Akateemikko (Academician). In 2008 he was awarded the international Antonio Feltrinelli Prize by Accademia Nazionale dei Lincei, as well as RIBA International Fellowship.