We consider models defined by a set of moment restrictions that may be subject to weak identification. Following the recent literature, the identification of the structural parameters is characterized by the Jacobian of the moment conditions. We unify several definitions of identification that have been used in the literature, and show how they are linked to the consistency and asymptotic normality of GMM estimators. We then develop two tests to assess the identification strength of the structural parameters in models that are (i) either linear or separable; (ii) neither linear nor separable. Both tests are straightforward to apply and allow to test specific subvectors without assuming identification of the components not under test. In simulations, our tests are well-behaved when compared to contenders, both in terms of size and power. In addition, we show how pretesting specific subvectors can improve inference by delivering shorter confidence regions with comparable coverage probability. Finally, when applied to the estimation of the Elasticity of Intertemporal Substitution, our test indicates weak instruments in a larger number of countries than tests proposed by Stock and Yogo (2005) and Montiel Olea and Pflueger (2012). Joint Eric Renault.