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Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The study is divided into three cases, depending on the topology of the surface. Case one is where the surface is a disk, in case two the surface is a planar domain (genus zero), andarXiv:math/0511740v1 fatcat:3ms335sswjfthajy43o6xhxogi