The idea of putting a shroud around a propeller is very old but it was only in 1934 that Stipa and Kort showed that a practical propeller of this design was possible. Kort designed a whole series of ducted propellers which proved very valuable when used in ships like tugs, pushboats, supply ships, trawlers etc. For this reason, his designs became known as Kort Nozzles.
His breakthrough was to shape the shroud around the propeller like an aerofoil rather than a simple ring. If the high-pressure side of the aerofoil faces outwards (so the top is like an aircraft wing right way up), the duct is of the decelerating type. If the low pressure side is outwards (so the top is like an aircraft wing upside down) the duct is accelerating. An accelerating duct produces positive thrust and increases the efficiency of heavily-loaded props (a heavily-loaded prop is one where the power being put through it is close to the maximum absorbable for that prop design). A decelerating duct produces a negative thrust and is valuable for reducing cavitation. A decelerating duct is a major contributor to noise reduction which is why it is getting more popular with modern warship designs.
The major problem with ducted props is flow separation. This causes the drag of the whole system to increase sharply, efficiency drops and the propeller tries to work in a highly irregular flow. This is bad. A mass of work has been done on the standards for ducted propellers and a number of "optimum" units designed. These are referred to by numbers, the simplest member being Nozzle No.19A. Nozzle No.22 for example, has a longer shroud in proportion to the prop diameter and is more efficient but gives poor backing characteristics. If a commercial designer wants to use Kort Nozzles, he'll decide on the characteristics he wants then selects the appropriate nozzle.
In summary, ducted propellers are well suited for situations where the propeller has to accommodate heavy loads. However, such prop designs cannot work safely without flanking rudders for the prop race since ducted props give virtually no backing control. In passing, flanking rudders are a pretty good idea since rudder control on ships doesn't usually work below around 10 knots and under this speed ships have to steer using their engines. While no great problem with twin and quad screw configurations, this is impossible with single screw ships. In this case, flanking rudders can offer some element of control at low speeds. In passing, German S-boats used a rudder trick (called the Lurssen Effekt) to push their sterns down and thus get slightly better efficiency for their props.
A pump jet is a developed Kort Nozzle in which the shroud is extended
and fixed guidevanes are installed fore and aft of the propeller. The duct
diameter increases from the entrance of the duct to the propeller so that
the velocity of the water falls and the pressure increases. This means
that the prop diameter is larger and the thrust loading drops. A lot of
the design art in this area is classified and the only treatment available
in open source comes from Henderson in 1963.