Superfluid
Superfluidity is a phase of matter or description of heat capacity in which unusual effects are observed when liquids, typically of helium-4 or helium-3, overcome friction by surface interaction when at a stage, known as the "lambda point" for helium-4, at which the liquid's viscosity becomes zero.  Also known as a major facet in the study of quantum hydrodynamics, it was discovered by Pyotr Kapitsa, John F. Allen, and Don Misener in 1937 and has been described through phenomenological and microscopic theories.

# Background
Although the phenomenologies of the superfluid states of helium-4 and helium-3 are very similar, the microscopic details of the transitions are very different.  Helium-4 atoms are bosons, and their superfluidity can be understood in terms of the Bose statistics that they obey.  Specifically, the superfluidity of helium-4 can be regarded as a consequence of Bose-Einstein condensation in an interacting system.  On the other hand, helium-3 atoms are fermions, and the superfluid transition in this system is described by a generalization of the BCS theory of superconductivity.  In it, Cooper pairing takes place between atoms rather than electrons, and the attractive interaction between them is mediated by spin fluctuations rather than phonons. See fermion condensate.  A unified description of superconductivity and superfluidity is possible in terms of gauge symmetry breaking.
Superfluids, such as supercooled helium-4, exhibit many unusual properties.  A superfluid acts as if it were a mixture of a normal component, with all the properties associated with normal fluid, and a superfluid component.  The superfluid component has zero viscosity, zero entropy, and infinite thermal conductivity.  (It is thus impossible to set up a temperature gradient in a superfluid, much as it is impossible to set up a voltage difference in a superconductor.)  One of the most spectacular results of these properties is known as the thermomechanical or "fountain effect".  If a capillary tube is placed into a bath of superfluid helium and then heated, even by shining a light on it, the superfluid helium will flow up through the tube and out the top as a result of the Clausius-Clapeyron relation.  A second unusual effect is that superfluid helium can form a layer, a single atom thick, up the sides of any container in which it is placed.
A more fundamental property than the disappearance of viscosity becomes visible if superfluid is placed in a rotating container.  Instead of rotating uniformly with the container, the rotating state consists of quantized vortices. That is, when the container is rotated at speed below the first critical velocity (related to the quantum numbers for the element in question) the  liquid remains perfectly stationary.  Once the first critical velocity is reached, the superfluid will very quickly begin spinning at the critical speed.  The speed is quantized - i.e. it can only spin at certain speeds.

# Applications
Recently in the field of chemistry, superfluid helium-4 has been successfully used in spectroscopic techniques, as a quantum solvent. Referred to as Superfluid Helium Droplet Spectroscopy (SHeDS), it is of great interest in studies of gas molecules, as a single molecule solvated in a superfluid medium allows a molecule to have effective rotational freedom - allowing it to behave exactly as it would in the "gas" phase.
Superfluids are also used in high-precision devices, such as gyroscopes, which allow the measurement of some theoretically predicted gravitational effects (for an example see the Gravity Probe B article).
Recently, superfluids have been used to trap light and slow its speed. In an experiment, performed by Lene Hau, light was passed through a superfluid and found to be slowed to 17 metres per second from its normal speed of 299,792,458 metres per second in vacuum.  This does not change the absolute value of c, nor is it completely new: any medium other than vacuum, such as water or glass, also slows down the propagation of light in a certain fraction.
The Infrared Astronomical Satellite (IRAS), launched in January 1983 to gather infrared data was cooled by 720 litres of superfluid helium, maintaining a temperature of 1.6K (-271.4 C).

# Recent discoveries
Physicists have recently been able to create a Fermionic condensate from pairs of ultra-cold fermionic atoms.  Under certain conditions, fermion pairs form diatomic molecules and undergo BoseEinstein condensation.  At the other limit, the fermions  (most notably superconducting electrons) form Cooper pairs which also exhibit superfluidity.  This recent work with ultra-cold atomic gases has allowed scientists to study the region in between these two extremes, known as the BEC-BCS crossover.
Additionally, supersolids might have also been discovered, in 2004, by physicists at Penn State University. When helium-4 is cooled, below about 200 mK under high pressures, a fraction (~1%) of the solid appears to become superfluid .