This is a new, completely revised, updated and enlarged edition of the author's Ergebnisse vol. 46: 'Spin Glasses: A Challenge for Mathematicians' in two volumes (this is the 2nd volume). In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called 'spin glasses'. These models are simple and rather canonical random structures, of considerable interest for several branches of science (statistical physics, neural networks and computer science). The physicists studied them by non-rigorous methods and predicted spectacular behaviors. This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics. The present Volume II contains a considerable amount of new material, in particular all the fundamental low-temperature results obtained after the publication of the first edition.