Abstract: Beginning with the solution of the classical Plateau problem—the problem of finding an area-minimizing disk whose boundary is a prescribed simple closed curve in Euclidean 3-space—we will survey some applications of Calculus of Variations to solve geometric extremal problems. Particular emphasis will be placed on the problem of finding a smooth surface in 3-space with minimum total squared mean curvature and prescribed isoperimetric ratio and genus. Such minimizers of the total squared mean curvature (also called the bending energy) were proposed in 1970 by Biologist Peter Canham to model the shape of red blood cells and lipid bilayers.