Math 227 is a topics course in topology, broadly defined. The topic changes from quarter to quarter. Three of the simplest types of infinite groups are free groups, surface groups and free abelian groups. This course focuses on the automorphism groups of finite graphs, closed orientable surfaces and n-dimensional tori. The approach will be to survey major results and to highlight some of the key underlying techniques, such as actions on R-trees and the construct of spaces such as outer space and the curve complex.