After a brief introduction to crystalline organic superconductors and metals, we shall describe two recently-observed exotic phases that occur only in high magnetic fields. The first involves measurements of the non-linear electrical resistance of single crystals of the charge-density-wave (CDW) system (Per)$_2$Au(mnt)$_2$ in static magnetic fields of up to 45 T and temperatures as low as 25 mK. The presence of a fully gapped CDW state with typical CDW electrodynamics at fields higher that the Pauli paramagnetic limit of 34 T suggests the existence of a modulated CDW phase analogous to the Fulde-Ferrell-Larkin-Ovchinnikov state. Secondly, measurements of the Hall potential of single crystals of $alpha$-(BEDT-TTF)$_2$KHg(SCN)$_4$, made using a variant of the Corbino geometry in quasistatic magnetic fields, show persistent current effects that are similar to those observed in conventional superconductors. The longevity of the currents, large Hall angle, flux quantization and confinement of the reactive component of the Hall potential to the edge of the sample are all consistent with the realization of a new state of matter in CDW systems with significant orbital quantization effects in strong magnetic fields.
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