This book investigates the close relation between quite sophisticated function spaces, the regularity of solutions of partial differential equations (PDEs) in these spaces and the link with the numerical solution of such PDEs. It consists of three parts. Part I, the introduction, provides a quick guide to function spaces and the general concepts needed. Part II is the heart of the monograph and deals with the regularity of solutions in Besov and fractional Sobolev spaces. In particular, it studies regularity estimates of PDEs of elliptic, parabolic and hyperbolic type on non smooth domains. Linear as well as nonlinear equations are considered and special attention is paid to PDEs of parabolic type. For the classes of PDEs investigated a justification is given for the use of adaptive numerical schemes.  Finally, the last part has a slightly different focus and is concerned with traces in several function spaces such as Besov- and Triebel-Lizorkin spaces, but also in quite general smoothness Morrey spaces.  The book is aimed at researchers and graduate students working in regularity theory of PDEs and function spaces, who are looking for a comprehensive treatment of the above listed topics.