The beautiful Riemann theorem states that a series can change its sum after permutation of the terms. Many brilliant mathematicians, among them P. Levy, E. Steinitz and J. Marcinkiewicz considered such effects for series in various spaces. In 1988, the authors published the book Rearrangements of Series in Banach Spaces. Interest in the subject has surged since then. In the past few years many of the problems described in that book - problems which had challenged mathematicians for decades - have in the meantime been solved. This changed the whole picture significantly. In the present book, the contemporary situation from the classical theorems up to new fundamental results, including those found by the authors, is presented. Complete proofs are given for all non-standard facts. The text contains many exercises and unsolved problems as well as an appendix about the similar problems in vector-valued Riemann integration. The book will be of use to graduate students and mathe- maticians interested in functional analysis.