In addition to the good information in @ZachLipton's answer, many aviation training companies offer simulator training for aircraft types other than those that most airlines fly, for example the Kodiak, Citation, or King Air.
This type of training is often used for training individuals flying their own advanced aircraft or for professional pilots. Pilots flying for corporate or charter outfits will typically do initial and recurrent training with such aviation training facilities.
Examples of such training facilities would be the Spokane Turbine Center, Flight Safety International, SIMCOM Aviation Training, and many others.
In addition to these companies that specialize in aviation training for advanced aircraft, many flight schools that specialize in primary flight training also include training in high level FTDs—which have also been used by some airlines, at least in the past. Examples of such schools would be MSU and UND, among many others.