In 1966, E.H. Lieb and D.C. r1attis published a book on "Mathematical Physics in One Dimension" [Academic Press, New York and London] which is much more than just a collection of reprints and which in fact marked the beginnings of the rapidly growing interest in one-dimensional problems and materials in the 1970's. In their Foreword, Lieb and r~attis made the observation that " ... there now exists a vast literature on this subject, albeit one which is not indexed under the topic "one dimension" in standard indexing journals and which is therefore hard to research ... ". Today, the situation is even worse, and we hope that these Proceedings will be a valuable guide to some of the main current areas of one-dimensional physics. From a theoretical point of view, one-dimensional problems have always been very attractive. Many non-trivial models are soluble in one dimension, while they are only approximately understood in three dimensions. Therefore, the corresponding exact solutions serve as a useful test of approximate ma?? thematical methods, and certain features of the one-dimensional solution re?? main relevant in higher dimensions. On the other hand, many important phe?? nomena are strongly enhanced, and many concepts show up especially clearly in one-dimensional or quasi -one-dimensional systems. Among them are the ef?? fects of fluctuations, of randomness, and of nonlinearity; a number of in?? teresting consequences are specific to one dimension.