  We review three different types of viscoelastic surface instabilities: The
Rayleigh -- Plateau, the Saffman -- Taylor and the Faraday instability. These
instabilities are classical examples of hydrodynamic surface instabilities. The
addition of a small amount of polymers to pure water can alter its flow
behavior drastically and the type of instability may change not only
quantitatively but also qualitatively. We will show that some of the observed
new phenomena can be explained by the use of simple rheological models that
contain most of the underlying physical mechanisms leading to the instability.
A quantitative description however is often only possible close to the onset of
the instability or for weak deviations from Newtonian behavior. A complete
theoretical description is still lacking when the system is driven far from
equilibrium or for fluids with strong non-Newtonian behavior.
