Graph Separators with Applications is devoted to techniques for obtaining upper  and lower bounds on the sizes of graph separators -- upper bounds being obtained  via decomposition algorithms. The book surveys the main approaches to obtaining  good graph separations, while the main focus of the book is on techniques for  deriving lower bounds on the sizes of graph separators. This asymmetry in focus  reflects our perception that the work on upper bounds, or algorithms, for graph  separation is much better represented in the standard theory literature than is  the work on lower bounds, which we perceive as being much more scattered  throughout the literature on application areas. Given the multitude of notions  of graph separator that have been developed and studied over the past (roughly)  three decades, there is a need for a central, theory-oriented repository for the  mass of results. The need is absolutely critical in the area of lower-bound  techniques for graph separators, since these techniques have virtually never  appeared in articles having the word `separator' or any of its near-synonyms in  the title. Graph Separators with Applications fills this need.