Topics in Modern Regularity Theory This book contains lecture notes of a series of courses on the regularity theory of partial differential equations and variational problems held in Pisa and Parma in the years 2009 and 2010. The contributors Nicola Fusco Tristan Rivière and Reiner Schätzle provide three updated and extensive introductions to various aspects of modern Regularity Theory concerning: mathematical modelling of thin films and related free discontinuity problems analysis of conformally invariant variational problems via conservation laws and the analysis of the Willmore functional. Each contribution begins with a very comprehensive introduction and is aimed to take the reader from the introductory aspects of the subject to the most recent developments of the theory. Ernst Kuwert and Reiner Schätzle The Willmore functional Tristan Rivière The Role of Conservation Laws in the Analysis of Conformally Invariant Problems B. De Maria and N. Fusco Equilibrium configurations of epitaxially strained elastic films.