In Galois theory one is interested in finding a polynomial over a field that has a given Galois group. A more desirable polynomial is one that parametrizes all such polynomials with that given group as its corresponding Galois group. These are called generic polynomials and we provide detailed proofs of two theorems that give methods for constructing such polynomials. Furthermore, we construct generic polynomials for Sn, C3, V , C4, C6, D3, D4, and D6