To \(x\in S_X\), the unit sphere of a Banach space \(X\), one associates its state space \(S_x=\{x^\ast\in S_{X^{\ast}}\mid x^\ast (x)=1\}\). \(x\) is a called a vertex if the span of \(S_x\) is weak-star dense in \(X^\ast\). A unitary is a vertex such that \(S_x =X^\ast\) and the set of unitaries is denoted by \(\mathcal{U}\). The authors remark that in \(C^\ast\)-algebras the notion of a vertex and a unitary coincide. The main objective in the paper is the generalization of the important concepts vertex and unitary from algebras to general Banach spaces. The elements of \(\mathcal{U}\) are localized by various concepts. It is, e.g., shown that \(x\in \mathcal{U}\subset X\) if and only if it is a unitary of \(X^{\ast\ast}\) and that \(\mathcal{U}\) is contained in the set of strongly extreme points of \(B_X\), a property vertices need not share. Let \(K\) be compact Hausdorff space and \(C(K,X)\) the \(X\)-valued continuous functions on \(K\), with sup-norm. \(\mathcal{U}\) in \(C(K,X)\) is studied, and characterized in the case \(X\) is a function algebra as the set of those \(f\) such that \(f(k)\) is a unitary in \(X\) for all \(k\in K\). It is also pointed out by an example that this is not true in general, not even for two-dimensional spaces \(X\). The authors end the paper by investigating how \(T\in\mathcal{L}(X)\) being a unitary is reflected in the position of \(T^\ast\) (note that if \(T^\ast\) is a unitary in \(\mathcal{L}(X^\ast)\), then \(T\) is a unitary in \(\mathcal{L}(X)\)). Whether \(T^\ast\) is always a unitary whenever \(T\) is, is left as an open question.