This is quite a big subject, so you will probably get it a bit at a time from different people.
There are two interpretations of 'printing in Adobe RGB'. One is that the printer (a physical object, or a company) can handle files tagged with Adobe RGB and another is that the gamut of the print will match the gamut of Adobe RGB. The latter isn't going to happen, the former could happen.
Here is an example that might help to clarify the issue. I expect that you will have more questions...
This is the envelope of the gamuts of Adobe, sRGB and the Epson 3800. The Epson 3800 is, in reality, a CMYK printer, with a fairly large gamut for a CMYK printer (not the greatest, but not the smallest). The 'envelope' means that the diagram shows the colours that are farthest from neutral, irrespective of the brightness of the colours. (It's really a top-down view of a three-dimensional graph)
You can see that Adobe RGB has a wider gamut than sRGB and that neither of them match the shape of the gamut of the 3800. The printer fails miserably in the pure reds, greens and blues (especially blues) but does quite well in the yellows and cyans. No surprise. It is using what is basically a CMYK ink set, and the C, M and Y inks are not perfect.
Now look at the gamuts in the darker colours. This is at a single brightness corresponding to L*=30, ie what would be perceived by a human as equal to a 30% grey (not 30% reflectance, but perceptually 30%).
Ignore the dots in this graph.
This shows that the 3800 is doing very well in most colours, but still failing miserably in the blues. Its gamut exceeds both Adobe RGB and sRGB in the green-cyan region.
Now look at the lighter colours, at L*=73.
Now the 3800 is closer to sRGB than Adobe RGB, but is still failing in the blues.
The printer cannot print the full gamut of either sRGB or Adobe RGB, but if it gets a file in Adobe RGB it has the
potential to print a wider gamut than if it gets sent an sRGB file.
The software handles the conversion between the printer's 'colour space' and the file's colour space - usually by converting to Lab first, then back to RGB for sending to the printer (just to confuse things, many inkjet printers are seen by the printing software as RGB devices when they are really CMYK)
So, in summary, the 3800 cannot print the Adobe RGB gamut, but the software controlling it can handle Adobe RGB files. It can't print the sRGB gamut either.
Good photo editing programs will show you if there are any colours outside the gamut of your display or printing device. In many cases there may not be a big problem. Here, for example, are the colours of a landscape photo (which is in Adobe RGB) shown for L*=53 (ie equivalent to about 20% reflectance). The cluster of dots represents the actual colours in use. Most of the outliers at the edges of the profiles are bugs in the viewing software (now fixed, I hope).
Note that most of the colours are within all three spaces, including the printers - the printer can print all the colours in this file. Had I used sRGB I would have lost some of the saturated yellow-greens, which are quite important in landscape photography. The image may be none the worse for losing them, however.
I'd just like to clarify something that Bifurcator wrote. Colour space is independent of bit depth. An 8-bit file in Adobe RGB has exactly the same gamut as a 16-bit file in Adobe RGB, the only difference is that the colours are described more coarsely in the 8-bit.
There are some large colour spaces that are best described in 16 bits because an 8-bit description would be too coarse, but the overall size of the colour space is not affected by how many bits per pixel per channel are used.
Raw files do not have a colour space assigned to them - they have the colour space native to the camera. When they are converted to an RGB format, they need to be converted to a colour space. You may lose colours when that happens. It may not matter to you. If you work on the Raw file in Lab (as Lightroom does) you won't lose colours until the conversion is made to an RGB space.
Good luck,
Helen