We study the local dual spaces of a Banach space X, which can be described as the subspaces of X* that have the properties that the principle of local reflexivity attributes to X as a subspace of X**. We give several characterizations of local dual spaces, which allow us to show many examples. Moreover, every separable space X has a separable local dual Z, and we can choose Z with the metric approximation property if X has it. We also show that a separable space containing no...