This book presents a comprehensive and coherent picture of how molecules diffuse across a liquid that is, on average, only two molecules thick. It begins by characterizing bilayers structurally, using X-ray diffraction, and then mechanically by measuring elastic moduli and mechanisms of failure. Emphasis is placed on the stability and mechanical properties of plant membranes that are subject to very large osmotic and thermal stresses. Using this information, the transport of molecules of increasing complexity across bilayers is analyzed.