We give a complete and detailed proof of Harer's stability theorem for the
homology of mapping class groups of surfaces, with the best stability range
presently known. This theorem and its proof have seen several improvements
since Harer's original proof in the mid-80's, and our purpose here is to
assemble these many additions.Comment: Proof of claim 3 in the spectral sequence argument corrected (see new
lemma 2.5, corollaries 2.6 and 2.7). To appear in the Handbook of Modul
