If \(B(K)\) denotes the space of all bounded, real-valued functions on a bounded subset \(K\) of a Banach space \(X\), the evaluation map \(E\) from \(X^*\) to \(B(K)\) is defined by: \(E(x^*)(k)= \langle x^*,k\rangle\) for all \(x^*\in X^*\) and \(k\in K\). If \(K\) denotes a bounded subset of \(X^*\) instead of \(X\), analogous evaluation maps from \(X\) and \(X^{**}\) to \(B(K)\) are similarly defined. In this paper, various properties of the set \(K\) which are related to compactness in some way or another (e.g. weak compactness, the Dunford-Pettis property) are characterized in terms of corresponding properties of these evaluation maps or their restrictions to subspaces. The paper brings together in a unified fashion numerious results of this nature which are scattered through the literature (and proved by widely different techniques).