This books presents modern functional analytic methods for the sensitivity analysis of some infinite dimensional sy- stems governed by partial differential equations. The main topics are treated in a general and systematic way including many classical applications such as the Signorini Problem, the elastic-plastic torsion problem and the visco-elastic- plastic problem. The "material derivative" is defined from which any kind of shape derivative of a cost functional can be derived. New results about the wave equation and the unilateral problem are also included in the book which will serve as a theoretical framework for the algorithmic appro- ach to shape optimization problems.