  In this chapter we give an overview of the properties of X-ray binary systems
containing a weakly magnetized neutron star. These are old (Giga-years
life-time) semi-detached binary systems containing a neutron star with a
relatively weak magnetic field (less than $\sim 10^{10}$ Gauss) and a low-mass
(less than $1 M_\odot$) companion star orbiting around the common center of
mass in a tight system, with orbital period usually less than 1 day. The
companion star usually fills its Roche lobe and transfers mass to the neutron
star through an accretion disk, where most of the initial potential energy of
the in-falling matter is released, reaching temperatures of tens of million
Kelvin degrees, and therefore emitting most of the energy in the X-ray band.
Their emission is characterized by a fast-time variability, possibly related to
the short timescales in the innermost part of the system. Because of the weak
magnetic field, the accretion flow can approach the neutron star until it is
accreted onto its surface sometimes producing spectacular explosions known as
type-I X-ray bursts. In some sources, the weak magnetic field of the neutron
star ($\sim 10^8-10^9$ Gauss) is strong enough to channel the accretion flow
onto the polar caps, modulating the X-ray emission and revealing the fast
rotation of the neutron star at millisecond periods. These systems are
important for studies of fundamental physics, and in particular for test of
Relativity and alternative theories of Gravity and for studies of the equation
of state of ultra-dense matter, which are among the most important goals of
modern physics and astrophysics.
