The domain with largest area for fixed circumference in plane is the disk. It has been conjectured that the minimal surface with largest area for fixed circumference in R^n is also the flat disk. I will survey the history of this open conjecture. In later seminars I will talk about the isoperimetric inequality for minimal surfaces in a Riemannian manifold (Part II) and about the joint work with R. Schoen in R^3 (Part III).