  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).
