Abstract: A compact subset K of C^n is said to be polynomially convex if every point in the complement of K can be separated from K by a polynomial. Oka-Weil theorem tells us that holomorphic functions can be approximated by polynomials on polynomially convex sets. This is a generalization of Runge approximation theorem to higher dimension. We will go through some approximation results in complex analysis, look at examples of polynomially convex sets, and analyze the local polynomial convexity of real n-dimensional manifolds embedded in C^n.