The theory of minimal submanifolds is a fascinating field in differential geometry. The simplest, one-dimensional minimal submanifold, the geodesic, has been studied quite exhaustively, yet there are still a lot of interesting open problems. In general, minimal submanifold theory deeply involves almost all major branches of mathematics; analysis, algebraic and differential topology, geometric measure theory, calculus of variations and partial differential equations, to name just a few of them.
In these lecture notes our aim is quite modest. We discuss minimal surfaces in R3 and concentrate on the class of the embedded complete minimal surfaces of finite topological type.