This book presents first of all the theory of linearly and nonlinearly elastic shells. It originated in the ISFMA-CIMPA School on “Differential Geometry: Theory and Applications“, which was held in August 2006 in the Chinese-French Institute for Applied Mathematics (ISFMA). The school was jointly organized by the ISFMA and the CIMPA (International Centre for Pure and Applied Mathematics), Nice, France. The book consists of four articles. The first article “An Introduction to Differential Geometry in R3” by P. G. Ciarlet develops systematically, and from the very beginning, differential geometry of three-dimensional Euclidean space. The second article “An Introduction to Shell Theory” by P. G. Ciarlet and C. Mardare presents three-dimensional theory of elastic bodies and then two-dimensional theory of elastic shells. The fundamental differential equations are derived here and their properties are studied. The last two articles, “Some New Results and Current Challenges in the Finite Element Analysis of Shells“ by D. Chapelle and “A Differential Geometry Approach to Mesh Generation“ by P. Frey, deal with numerical methods suitable for solutions of the above mentioned fundamental equations. We find here the notion of a finite shell element within the framework of finite element methods and mesh generation for the finite element method. Though the exposition starts from the very beginning it leads to the most recent procedures. The book is very important for specialists in the field. On the other hand, because the presentation is very systematic and rather self-contained, it is also convenient for graduate students.