We give a survey on invariant theory of finite group actions on polynomial rings, from classical results of Gauss, Hilbert, Noether, Shephard, Todd, Chevalleyto modern approaches of Stanley, Wanatabe,  Auslander and Steinberg using homological algebra, algebraic geometry and combinatorics. We will briefly introduce noncommutative invariant theory, based on some recent work of Jorgensen, Jing, Kirkman, Kuzmanovich, Bao, He, Qin. If time permits, we will also propose some open questions and future directions.

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