Varifolds are measures that describe generalized surfaces in geometric measure theory. Due to their compactness properties, varifolds provide a favorable framework to study geometric variational problems with the Direct Method. An example of a geometric energy functional that also involves curvature is the Canham-Helfrich energy. Its minimizers model the equilibrium configurations of biological membranes, like the famous biconcave shape of human red blood cells. After a gentle introduction to the theory of curvature varifolds, I present an existence result for multiphase membranes, that I obtained in collaboration with Luca Lussardi (Torino) and Ulisse Stefanelli (Vienna).