Lipid bilayers constitute the membrane that encloses every animal cell and many of its interior structures, including the nuclear envelope, the organelles and the endoplasmic reticulum. They possess some of the features of conventional engineering shell structures such as flexural resistance, but they are unusual in that they also possess the properties of two-dimensional curved fluid sheets. This combination is responsible for a myriad of remarkable mechanical and physical responses that make lipid bilayers a unique and fascinating topic of study. The basic molecular mechanism responsible for this range of behavior is the lipid molecule. It is polar, with one end – the molecular tail - being hydrophobic, and the other – the head group - hydrophilic. The lipids constitute the two leaves of the bilayer, with the hydrophobic tails juxtaposed in such a way as to exclude the surrounding aqueous solution. Thus the bilayer is the product of a self-assembly process driven by a clear physical mechanism. In recent years the study of this subject has been undertaken by experts in physics and mechanics. The perspective brought to bear by the mechanics community has facilitated a large number of significant advances, not only with respect to an improved understanding of the foundations of the subject, but also with respect to the modeling of phenomena that had heretofore been treated on an ad hoc basis without the benefit of the overall intellectual viewpoint that modern mechanics brings to bear. Thus the broad framework afforded by mechanics has led to important conceptual advances in such topics as phase equilibria in lipid bilayers, diffusion and transport phenomena, cell adhesion and motility, tubule formation, coupled electromechanical response, edge and pore effects, intra-and extra- membrane viscous flow, inter-leaf friction, the role of tilt (known to the shell theorist as transverse shear deformation), bifurcation and instability, membrane rheology, models based on molecular considerations, and so on. Indeed the field is nowadays extremely active, and has grown into a major discipline lying at the intersections of mechanics, bio-physics and applied mathematics. One of the most important conceptual aspects of the subject is the profound interplay it exhibits between mechanics and geometry. Thus the field makes essential use of advanced topics in the differential geometry of surfaces. Indeed it requires considerable facility with the latter subject, and provides an opportunity to exercise our knowledge of virtually that entire branch of applied mathematics. For this reason much modern work on bilayers is primarily geometric in nature. The presentations will be carefully crafted overviews of the basic theory and its various enhancements and extensions from several points of view. They will also include surveys of relevant differential geometry and variational methods that are essential to a proper understanding, overviews on the foundations of the subject from various perspectives, applications of modern bifurcation theory to the analysis of membrane equilibria, and further developments encompassing a range of coupled-field phenomena. The course is addressed to doctoral students, post-doctoral researchers and academics interested in the use of mechanics to model, analyze and understand the physics of lipid bilayers. Lipid bilayers are ubiquitous in biology, and related structures occur in fluid interfaces and experimental vesicles. While the course is largely self-contained, students would benefit from prior exposure to courses on continuum mechanics, tensor analysis and elementary electromagnetism.