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Five and a half roads to form a millisecond pulsar

In this review I discuss the characteristics and the formation of all classes of millisecond pulsars (MSPs). The main focus is on the stellar astrophysics of X-ray binaries leading to the production of fully recycled MSPs with white dwarf (WD) or substellar semi-degenerate companions. Depending on the nature of the companion star MSPs are believed to form from either low-mass X-ray binaries (LMXBs) or intermediate-mass X-ray binaries (IMXBs). For each of these two classes of X-ray binaries the evolutionary status of the donor star -- or equivalently, the orbital period -- at the onset of the Roche-lobe overflow (RLO) is the determining factor for the outcome of the mass-transfer phase and thus the nature of the MSP formed. Furthermore, the formation of binary MSPs is discussed in context of the (P,P_dot)-diagram, as well as new interpretations of the Corbet diagram. Finally, I present new models of Case A RLO of IMXBs in order to reproduce the two solar mass pulsar PSR J1614-2230.