This volume presents a fairly self-contained theory of certain singular coverings of toposes, including branched coverings.This is a field that should be of interest to topologists working in knot theory, as well as also to certain categorists. An unusual feature which distinguishes this book from classical treatments of the subject is an unexpected connection with a topic from functional analysis, namely, distributions. Although primarily aimed at topos theorists, this book may also be used as a textbook for advanced graduate courses introducing topos theory with an emphasis on geometric applications.