Neutron stars in low-mass X-ray binaries (NS-LMXBs) are unique laboratories of accretion physics, strong gravity and ultra-dense matter. I will give an overview of what we have learned in recent years by studying accretion flows and thermonuclear bursts in these systems.

I will first present and discuss the main result of a systematic study of their different accretion states: the discovery of a correlation between luminosity and spectral hardness. I will also show ongoing work on the connection between active (1-100% of the Eddington luminosity) and quiescent (down to 10^-6 times Eddington) phases of NS-LMXBs.

In the second part I will focus on the relation between mass accretion rate and the recurrence time of thermonuclear bursts (explosive nuclear burning on the neutron star surface), presenting results at the lowest and highest mass accretion rates. In particular, I will argue that rotation plays a larger role than we thought in setting the nuclear burning regimes on an accreting neutron star.