These notes consist of two parts: Selected Topics in Geometry, New York University 1946, 1) Notes by Peter Lax 2) Lectures on Differential Geometry in the Large, Stanford University 1956, Notes by J.W. Gray They are reproduced here with no essent ial change Heinz Hopf was a mathematician who recognized important mathema tical ideas and new mathematical phenomena through special cases. In the simplest background the central idea or the difficulty of a problem usually becomes crystal lear. Doing gecmetry in this fashion is joy. Hopf's great insight allows this approach to lead to serious ma- thematics, for most of the topics in these notes have become the star- ting-points of important further developments.