Mitsubishi Mr. Slim is the most common example of such a system. I have seen them in several places out west, but not so much in the eastern US. They are very popular in Japan, and other Pacific countries (notice the band names of most of the systems), but they are just starting to become common in the US.
As I understand it, these systems became popular in urban settings because of high-rise buildings. It is easier to route coolant lines down from the rooftop instead of massive central air ducts.
If you are retrofitting a house that doesn't have existing ductwork, these mini-split systems can be a preferable option. As any HVAC contactor can tell you, designing ducts for proper airflow can be a difficult task, and it can be almost impossible to get right in existing construction.
If you get one of the multi-zone ductless systems, you have a thermostat at each indoor unit, which can allow for more flexibility in setting different rooms to different temperatures.
They generally have very high efficiency ratings. In many cases, they have a higher SEER/EER than an equivalent centralized AC system.
The drawbacks to these systems are:
You have the indoor unit sticking out from the wall.
Since they are less popular, it can be harder to find a competant service company.
Fewer companies selling/servicing mean it can be more expensive to get work done.
The residental systems I have seen can support at most 3 indoor units. So, if you want good airflow in every room of your house, you're not going to get it.
The residential systems I have seen max out at ~3.5 tons of total cooling. So, a single system may be insufficient for a big (or poorly insulated) house.
I have a relative who converted his screened-in patio into a sun room. He then added a Mr. Slim to provide A/C. It's quiet and keeps the room cool. And since it is independent from the main house HVAC, the whole house doesn't need the A/C running constantly. This seems to be a really good fit.
At my office, we have one of these units installed in our server room at the office to provide cooling. We've had several problems, but I think the unit was undersized for the application. It's also a residental unit that wasn't made to cool a server room 24/7/365. It's quiet though.
I don't have any first hand experience with the heat pump version of the unit, so I am not sure how well they do in a heating scenario.