A heat-pump water heater 'radiator'.
A heat-pump water heater (HPWH) has a "radiator" similar to a car, but instead of hot water there's a refrigerant much colder than the outside air.
When outside air is pushed through the fins of the "radiator", it warms the cold refrigerant inside.
This warmed refrigerant is then compressed (which increases its temperature further) and fed though pipes that are in contact with the hot water. This warms the hot water and cools the refrigerant - and when this is returned to the "radiator" it expands and cools further.
Circulating refrigerant around like this shifts heat from the outside air into the hot water. A well-designed HPWH will use much less electricity than if you directly heated the water with an electric element.
Two types
There are two types of HPWH: they work the same way but look different. We tested both types.
Integral
Integral models have the heat pump and the water tank combined into a single unit. The heat pump sits on the top of the tank and the whole unit is installed outdoors, preferably near the house's bathroom.
When you buy an integral model you replace your existing cylinder. (This also has the advantage of freeing up your hot-water-cupboard space in the house.)
The Stiebel Eltron WWK 300, the Quantum 270-11AC3-134 and the Rheem 55131007/05 were all integral models.
Split
Split models use an outdoor heat pump unit, which looks similar to a space-heating heat pump. A small pump then circulates water between the outdoor heat-pump unit and the hot-water cylinder. There are limits on the length of the piping from the heat pump to the water cylinder, but the shorter distance the better.
Split models can use your existing hot-water cylinder, although you might wish to replace it if it's an older model.
The Econergy HP4000 LT and the National RS 62P5A/T were split models.
Future developments
The current HPWHs use standard heat-pump technology, similar to the heat pumps many people use to heat their homes.
But in Japan there are units in production using carbon dioxide as the refrigerant. These units have efficiencies roughly twice those of the models being sold here. It seems likely these energy-efficient models will find their way here eventually. We don't know when they'll arrive or what they'll cost - but we're watching.

