This volume constitutes the Proceedings of the Fourth Franco–Japanese Symposium on Singularities held at Toyama in August 2007 and also the Workshop on Singularities held at Niigata prior to this Symposium. Recently the research on singularities is widely expanding and is now applied in various areas in Mathematics and other sciences. Experts of singularities from many different fields are contributing their articles, mostly on original results and some surveys. The reader will benefit of knowing the vividly developing domains and will be inspired by many different approaches to singularities.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America