Total cross sections for Σ − and π − on beryllium, carbon, polyethylene and copper as well as total cross sections for protons on beryllium and carbon have been measured in a broad momentum range around 600 GeV/c. These measurements were performed with a transmission technique adapted to the SELEX hyperon-beam experiment at Fermilab. We report on results obtained for hadron-nucleus cross sections and on results for σ tot (Σ − N) and σ tot (π − N), which were deduced from nuclear cross sections.