  Since the advent of powerful new X-ray observatories, NASA's Rossi X-ray
Timing Explorer (RXTE), the Italian - Dutch BeppoSAX mission, XMM-Newton and
Chandra, a number of entirely new phenomena associated with thermonuclear
burning on neutron stars have been discovered. These include: (i) the discovery
of millisecond (300 - 600 Hz) oscillations during bursts, so called ``burst
oscillations'', (ii) a new regime of nuclear burning on neutron stars which
manifests itself through the generation of hours long flares about once a
decade, now referred to as ``superbursts'',(iii) discoveries of bursts from low
accretion rate neutron stars, and (iv) new evidence for discrete spectral
features from bursting neutron stars. In this article we review our current
understanding of thermonuclear bursts on neutron stars, with a focus on these
new phenomena.
