The AdobeRGB-standard dates back to 1998 and was originally designed to resemble the colour space of CMYK-prints as they are made in printing companies. It covers about half of the CIELAB colour space and shows considerably more saturation in the green and cyan tints, not coincidentally the tints where the human eye can perceive more than in for example the blue part of the spectrum.
For displaying this wider spectrum another technique is necessary compared with standard sRGB-monitors. The latter use - nowadays - white led-lighting. This can supply a relatively small spectrum of 'white', which is due to the fact that these are actually blue leds covered by a yellow phosphor layer. This is also the most important limitation for the actual range. Previously - up to about 2012 - AdobeRGB-monitors used ccfl-lighting because of this, which was phased out for regular monitors.
Right now this is also phased out for AdobeRGB-monitors and is replaced by led-lighting with an extra colour component. Initially there was experimentation with rgb led-lighting, but this was very expensive. Another disadvantage of this was that all leds of different colours age in a different manner, which caused problems after a while. Modern monitors use leds in multiple colours, with a coating in a third colour. For example, in the case of LG, GB-r: green and blue leds, covered in a red phosphor layer. A recent example of this is the Dell UltraSharp UP2716D. Another application of this is used by AU Optronics, who use RB-g leds: a mix of red and blue leds with green phosphor layer.
The most modern variant is a so-called quantum dots enhancement film. This is a layer with phosphor on nanoscale, which is placed between the (blue) backlight and the lcd-matrix. This technique is based on the same principle as quantum dots in televisions, only there the quantum dots are placed directly on the leds. The effect is the same: the width of the spectrum increases considerably, and with that the colour range. The big advantage of quantum dots on a seperate layer - or placed on some sort of rail over the led-lighting that is usually on the edge of a display - is that it does not require any major changes to the existing production lines.
All technical implementations of the hardware are one thing, on the software-side AdobeRGB - and other 'big' colour spaces - have just as many caveats. The most basic one is this: if software is not configured correctly to use a wider colour space, or cannot work with different colour profiles, the rendering of an AdobeRGB-monitor is incorrect. Colours are shown over saturated. This begins with something as simple as a browser: while Microsoft, Mozilla and Google are working on it, official support for wider colour spaces is still absent or does not work immaculately. In practice this means that you have to change between different modes with an AdobeRGB-monitor, depending on the application that you are working with. If you think that you are rewarded with a 'more beautiful colour quality' after connecting a monitor with a wider colour space, you are completely wrong. For now these kinds of monitors are primarily useful for those working with applications that can utilize the extra functionality. This is something to bear in mind, especially because these monitors are clearly more expensive than monitors with a less extensive colour space.