Quotients by algebraic group actions

We present different concepts of quotients including rational, categorical, geometric and good quotients. Starting with the set-theoretical approach and Rosenlicht’s theorem, we come to the quotient of a reductive group action on an affine variety along with possible generalizations. We discuss Mumford’s construction of the quotient of the set of semistable points defined by a $G$-linearized line bundle and variations of GIT quotients. We explain how quotient constructions motivate the study of certain extensions of the category of algebraic varieties.