In this thesis, our aim is to first layout the framework of optimal transport, and then demonstrate its usefulness in proving functional inequalities. This approach is advantageous as it provides extremely efficient proofs, while requiring potentially less work than when using a classical approach. In particular, the optimal transport approach grants the ability to prove inequalities with sharp constants and pinpoints conditions for which they hold with equality. For example, in the case of the Gagliardo-Nirenberg-Sobolev (GNS) Inequality, optimal transport recovers the inequality for arbitrary norms in $\mathbb{R}^n$.