Shape optimization and spectral theory „Shape optimization and spectral theory” is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization.It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet Robin and Steklov). This includes: existence of optimal shapes their regularity the case of special domains like triangles isospectrality quantitative form of the isoperimetric inequalities optimal partitions universal inequalities and numerical results.Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributorsAntunes Pedro R.S. Ashbaugh Mark Bonnaillie Noël Virginie Brasco Lorenzo Bucur Dorin Buttazzo Giuseppe  De Philippis Guido  Freitas Pedro  Girouard Alexandre  Helffer Bernard  Kennedy James Lamboley Jimmy  Laugesen Richard S.  Oudet Edouard  Pierre Michel  Polterovich Iosif  Siudeja Bartłomiej A. Velichkov Bozhidar