Contact binaries are close binary systems in which both components fill their
inner Roche lobes so that the stars are in direct contact, and in potential
mass and energy exchange. The most common such systems of low mass are the
so-called W UMa-type. In the last few years, there has been a growing
interest of the astronomical community in stellar mergers, primarily due to
the detection of gravitational waves (mergers of black holes and neutron
stars), but also because of an alternative model for the type Ia supernovae
(merger of two white dwarfs), which are again particularly important in
cosmology where they played a significant role in the discovery of dark
energy and the accelerated expansion of the Universe. In that sense, contact
systems of W UMa type with extremely low mass ratio are especially
interesting because there are indications that, in their case too, stars can
merge and possibly form fast-rotating stars such as FC Com stars and the
blue-stragglers, and (luminous) red novae such as V1309 Sco. Namely, the
previous theoretical research has shown that in the cases when the orbital
angular momentum of the system is only about three times larger than the
rotational angular momentum of the primary, a tidal Darwin's instability
occurs, the components can no longer remain in synchronous rotation, orbit
continue to shrink fast, and they finally merge into a single star. The above
stability condition for contact systems can be linked to a specific critical
mass ratio below which we expect a system to be unstable. We give an
overview of this condition and show how it can be used to identify potential
mergers. Finally, we discuss a number of known extreme mass ratio binaries
from the literature and consider prospects for future research on this
topic.