The L?vy Laplacian is an infinite-dimensional generalization of the well-known classical Laplacian. The theory has become well developed in recent years and this 2005 book was the first systematic treatment of the L?vy-Laplace operator. The book describes the infinite-dimensional analogues of finite-dimensional results, and more especially those features which appear only in the generalized context. It develops a theory of operators generated by the L?vy Laplacian and the symmetrized L?vy Laplacian, as well as a theory of linear and nonlinear equations involving it. There are many problems leading to equations with L?vy Laplacians and to L?vy-Laplace operators, for example superconductivity theory, the theory of control systems, the Gauss random field theory, and the Yang-Mills equation. The book is complemented by an exhaustive bibliography. The result is a work that will be valued by those working in functional analysis, partial differential equations and probability theory.