This book gives the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory in a self-contained and accessible manner. Although the field is often considered a classical one, it has recently been rejuvenated, thanks to the manifold applications where it plays an essential role. The book presents some important applications to shells, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and mesh generation in finite element methods. This volume will be very useful to graduate students and researchers in pure and applied mathematics. Contents: An Introduction to Differential Geometry in 3 (P G Ciarlet); An Introduction to Shell Theory (P G Ciarlet & C Mardare); Some New Results and Current Challenges in the Finite Element Analysis of Shells (D Chapelle); A Differential Geometry Approach to Mesh Generation (P Frey).