Topology and geometry have become useful tools in many areas of physics and engineering, and of course permeate every corner of research in today's mathematics. They often help us make fresh progress precisely because they are very unlike, and complement, traditional differential-equation-based methods. Yet they are almost never taught to students outside advanced pure mathematics. This course teaches a minimal amount of topology and geometry of maximal usefulness in applications, relying on pictures and avoiding abstract algebraic machinery. The motto of the course is: look at the generic situation, spot invariants, solve your problem by deformation. I hope that the course will train you to THINK IN PICTURES. Note: Some of you may have studied point-set topology (metric and topological spaces, continuous maps, compactness, etc.). The content of this course is different: it is usually called algebraic and differential topology. This course is designed so that familiarity with point-set topology is unnecessary.