We develop a unified approach to characterize approximation properties defined by spaces of operators. Our main result describes them in terms of the approximability of weak*-weak continuous operators. In particular, we prove that if $\mathcal{A}$ and $\mathcal{B}$ are operator ideals satisfying $\mathcal{A}\circ\mathcal{B}^{*}\subset\mathcal{K}$ , then the $\mathcal{A}(X,X)$ -approximation property of a Banach space X is equivalent to the following “metric” condition: for every Banach space Y and for every operator $T\in\mathcal{B}^{*}(Y,X)$ , there exists a net $(S_{\alpha})\subset\mathcal{A}(X,X)$ such that supα‖SαT‖≤‖T‖ and T*Sα*→T* in the strong operator topology on $\mathcal{L}(X^{\ast},Y^{\ast})$ . As application, approximation properties of dual spaces and weak metric approximation properties are studied.