In this talk, we will introduce the basics of invariant theory. We first define group actions and explore some basic group actions on the vector space R^n, including those by dihedral and symmetric groups. We then extend these actions to the so-called symmetric algebra on R^n. It is here that we look for invariants, the elements that are fixed by the entire group. Eventually we will want to stop writing down invariants, and instead characterize the full set of invariants. If time permits, we may discuss how the same groups act on the dual space of R^n as well as the derivations on R^n.