They have been long known as "Turkish toilets", and they are probably the most common type of toilet in the world.
They are the only type of toilet to be found at my school, and they are also the norm at older train stations, public parks, and countless other places. I don't prefer one over the other, having lived and worked in places where there were no toilets whatsoever.
My toilet at home is a very nice sit-down model which is equipped with a "washlet". It has a heated seat and a built in bidet, which can be adjusted for water temperature, pressure, and spray pattern. Pretty trick.