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We review the role in physics of the Pin groups, double covers of the fullLorentz group. Pin(1,3) is to O(1,3) what Spin(1,3) is to SO(1,3). Theexistence of two Pin groups offers a classification of fermions based on theirproperties under space or time reversal finer than the classification based ontheir properties under orientation preserving Lorentz transformations --provided one can design experiments that distinguish the two types of fermions.Many promising experimental setups give, for one reason or another, identicalresults for both types of fermions. Two notable positive results show that theexistence of two Pin groups is relevant to physics:  1) In a neutrinoless double beta decay, the neutrino emitted and reabsorbedin the course of the interaction can only be described in terms of Pin(3,1).  2) If a space is topologically nontrivial, the vacuum expectation values ofFermi currents defined on this space can be totally different when described interms of Pin(1,3) and Pin(3,1).  Possibly more important than the two above predictions, the Pin groupsprovide a simple framework for the study of fermions; they make possible cleardefinitions of intrinsic parities and time reversal. A section on Pin groups inarbitrary spacetime dimensions is included.Comment: 109 pages, 14 figure

THE PIN GROUPS IN PHYSICS: C, P AND T