In materials with strong local Coulomb interactions, simple defects such as atomic substitutions strongly affect both macroscopic and local properties of the system. A nonmagnetic impurity, for instance, is seen to induce magnetism nearby. Even without disorder, models of such correlated systems are generally not soluble in two or three dimensions, and so few exact results are known for the properties of such impurities. Nevertheless, some simple physical ideas have emerged from experiments and approximate theories. Here the authors review what we can learn about this problem from one-dimensional (1D) antiferromagnetically correlated systems. Experiments on the high-$T_c$ cuprate normal state which probe the effect of impurities on local charge and spin degrees of freedom are discussed, and compared with theories of single impurities in correlated hosts, as well as phenomenological effective Kondo descriptions. Subsequently, theories of impurities in $d$-wave superconductors including residual quasiparticle interactions are reviewed and compared with experiments in the superconducting state. Existing data exhibit a remarkable similarity to impurity-induced magnetism in the 1D case, implying the importance of electronic correlations for the understanding of these phenomena, and suggesting that impurities may provide excellent probes of the still poorly understood ground state of the cuprates.

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