Recent transport experiments in the original cuprate high temperature superconductor, La_2-xBa_xCuO_4, have revealed a remarkable sequence of transitions and crossovers which give rise to a form of dynamical dimensional reduction, in which a bulk crystal becomes essentially superconducting in two directions while it remains poorly metallic in the third. We identify these phenomena as arising from a distinct new superconducting state, the "striped superconductor," in which the superconducting

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... er is spatially modulated, so that its volume average value is zero. Here, in addition to outlining the salient experimental findings, we sketch the order parameter theory of the state, stressing some of the ways in which a striped superconductor differs fundamentally from an ordinary (uniform) superconductor, especially concerning its response to quenched randomness. We also present the results of DMRG calculations on a model of interacting electrons in which sign oscillations of the superconducting order are established. Finally, we speculate concerning the relevance of this state to experiments in other cuprates, including recent optical studies of La_2-xSr_xCuO_4 in a magnetic field, neutron scattering experiments in underdoped YBa_2Cu_3O_6+x, and a host of anomalies seen in STM and ARPES studies of Bi_2Sr_2CaCu_2O_8+δ .