Current research into the morphology of lipid membranes focuses on three areas: first, the transformations and fluctuations of the shape of freely suspended vesicles; second, the morphology of membranes that experience mutual interactions or external forces arising, for example, from a macroscopic surface or from optical tweezers; and third, the behavior of heterogeneous membranes with respect to bilayer asymmetry, intramembrane domains and anchored polymers. Our understanding of the morphology of lipid membranes has progressed substantially through the development of experimental techniques and theories.