Let X be a Banach space and Y be a closed subspace of X. Given a bounded linear functional f on Y , the Hahn-Banach theorem guarantees that there exists a linear extension ˜ f 2 X of f which preserves the norm of f. But it does not state that such ˜ f is unique or not. If every f in Y does have a unique norm preserving extension ˜ f in X , we say that Y has the unique extension property, or, following P. R. Phelps, the property U in X. A. E. Taylor [17] and S. R. Foguel [7] had shown that every subspace Y of X has the unique norm-preserving extension property in X if and only if the dual space X is strictly convex. As known in [11], X is smooth if X is strictly convex. The converse does not hold in general unless X is reflexive. In this thesis, we show that if a subspace Y of a Banach space X has the unique extension property then the norm of Y is outward smooth in X. The converse holds when Y is reflexive. Note that our conditions are local, i.e., they depend on Y only, but not on X. Several related results are also derived. Our work extends and unifies recent results in literature.