This book deals with the acceleration and storage of polarized proton beams in cyclic accelerators. Polarized proton beams with hundreds of GeV are, for example, essential for resolving the “spin crisis,” i.e., for understanding the angular momentum distribution inside nucleons. Experience with the Relativistic Heavy Ion Collider at the Brookhaven NationalLaboratory(RHICatBNL)hasshownthatpolarizationat205GeV can be obtained, and tests at the pre-accelerating Alternating Gradient S- chrotron(AGS)haveshownthatatleastupto24GeV,thebeampolarization canbequiteundisturbedwhentheacceleratoriswelladjusted,exceptats- cial energies where resonances occur. In particular, it has not been necessary to study closely the variation of the protons’ spin directions across the phase space of the beam. However, at very high energies such as in RHIC, TEVA- TRON, HERA-p, LHC, or a future VLHC, new phenomena can occur that can lead to a signi?cantly diminished beam polarization. For these high energies, it is necessary to look in more detail at the spin motion at each point in phase space, and for this the invariant spin ?eld provestobeausefultool.This?eldgivesrisetoanadiabaticinvariantofsp- orbit motion, and it de?nes the maximum time-average polarization that is usable in a particle physics experiment. Furthermore, the invariant spin ?eld allows the amplitude dependent spin tune to be de?ned and computed and thereby opens the way to a clear evaluation of the e?ects of higher-order spin orbit resonances. In particular, the strengths and the depolarizing e?ects of these resonances can only be determined once the amplitude-dependent spin tune has been computed.