For anyone unfamiliar with the Thorium fission cycle, the first five minutes of this Creative Commons film give a good overview with a more detailed exploration of the topic later in the film:
Yes. Thorium is the yummy chocolate and sprinkles donuts version of todays nuclear reactor. All the benefits and almost none of the drawbacks.
You're talking specifically about liquid fluoride thorium reactors. Other types of thorium (or thorium-uranium mix) reactors exist, and are in commercial use now. Canada's CANDU reactors, for example, can run on thorium.
They don't exist, do they?
Not at serious production scale yet, no. There have been molten salt reactors, and there have been thorium reactors of various types, and many commercially-operating reactors now are capable of running wholly or partially on thorium, and pebble bed reactors can run on thorium fuel if you're clever with the pebble handling (or willing to mix in uranium), but at the moment the nuclear plants that are ready to build are primarily pressurized light water reactors running on lightly enriched uranium.