Accreting pulsars: mixing-up accretion phases in transitional systems

In the last 20 years our understanding of the millisecond pulsar (MSP) population changed dramatically. Thanks to RXTE, we discovered that neutron stars in LMXBs spins at 200-750 Hz frequencies, and indirectly confirmed the recycling scenario, according to which neutron stars are spun up to ms periods during the LMXB-phase. In the meantime, the continuous discovery of rotation-powered MSPs in binary systems in the radio and gamma-ray band (mainly with the Fermi LAT) allowed us to classify these sources into two "spiders" populations, depending on the mass of their companion stars: Black Widow, with very low-mass companion stars, and Redbacks, with larger companions possibly filling their Roche lobes but without accretion. It was soon regained that MSPs in short orbital period LMXBs are the progenitors of the spider populations of rotation-powered MSPs, although a direct link between accretion- and rotation-powered MSPs was still missing. In 2013 XMM-Newton spotted the X-ray outburst of a new accreting MSP (IGR J18245-2452) in a source that was previously classified as a radio MSP. Follow up observations of the source when it went back to X-ray quiescence showed that it was able to swing between accretion- to rotation-powered pulsations in a relatively short timescale (few days), promoting this source as the direct link between the LMXB and the radio MSP phases. Following discoveries showed that there exists a bunch of sources, which alternates X-ray activity phases, showing X-ray pulsations, to radio-loud phases, showing radio pulsations, establishing a new class of MSPs: the Transitional MSP. In this review we describe these exciting discoveries and the properties of accreting and transitional MSPs, highlighting what we know and what we have still to learn about in order to fully understand the (sometime puzzling) behavior of these systems and their evolutive connection (abridged).