CAMPANA, SergioSergioCAMPANA0000-0001-6278-1576Di Salvo, TizianaTizianaDi Salvo2020-10-072020-10-072018978-3-319-97615-00067-0057http://hdl.handle.net/20.500.12386/27655In the last 20 years our understanding of the millisecond pulsar population changed dramatically. Thanks to the large effective area and good time resolution of the NASA X-ray observatory Rossi X-ray Timing Explorer, we discovered that neutron stars in Low Mass X-ray Binaries (LMXBs) spins at frequencies between 200 and 750 Hz, and indirectly confirmed the recycling scenario, according to which neutron stars are spun up to millisecond periods during the LMXB-phase. In the meantime, the continuous discovery of rotation-powered millisecond pulsars in binary systems in the radio and gamma-ray band (mainly with the Fermi Large Area Telescope) allowed us to classify these sources into two "spiders" populations, depending on the mass of their companion stars: Black Widow pulsars, with very low-mass companion stars, and Redbacks, with larger mass companion stars possibly filling their Roche lobes without accretion of matter onto the neutron star. It was soon regained that millisecond pulsars in short orbital period LMXBs are the progenitors of the spider populations of rotation-powered millisecond pulsars, although a direct link between accretion-powered and rotation-powered millisecond pulsars was still missing. In 2013 the ESA X-ray observatory XMM-Newton spotted the X-ray outburst of a new accreting millisecond pulsar (IGR J18245-2452) in a source that was previously classified as a radio millisecond pulsar, probably of the Redback type. Follow up observations of the source when it went back to X-ray quiescence showed that it was able to swing between accretion-powered to rotation-powered pulsations in a relatively short timescale (few days), promoting this source as the direct link between the LMXB and the radio millisecond pulsar phases. Following discoveries showed that there exists a bunch of sources which alternates X-ray activity phases, showing X-ray coherent pulsations, to radio-loud phases, showing radio pulsations, establishing a new class of millisecond pulsars, the so-called transitional millisecond pulsars. In this review we describe these exciting discoveries and the properties of accreting and transitional millisecond pulsars, highlighting what we know and what we have still to learn about in order to fully understand the (sometime puzzling) behaviour of these systems and their evolutive connection.STAMPAenBook part10.1007/978-3-319-97616-7_4https://link.springer.com/chapter/10.1007%2F978-3-319-97616-7_42018ASSL..457..149CFIS/05 - ASTRONOMIA E ASTROFISICAERC sectors::Physical Sciences and Engineering::PE9 Universe sciences: astro-physics/chemistry/biology; solar systems; stellar, galactic and extragalactic astronomy, planetary systems, cosmology, space science, instrumentation