Admittedly you have to be at least 40 to remember the Dutch DAF car (it was absorbed into Volvo in the 1970s). It pioneered a transmission using a rubber pulley rather than individual gears.
The "rubber band" transmission later mutated into a steel belt and was tried in various small Fiats and Fords, but without much success. However, it is now catching on in the USA, of all places. In 2005 1% of cars in America had CVT (for Constantly Variable Transmission), By 2010 it was 7%, and according to the forecasting company IHS Automotive, it will hit 16% by 2016.
The advantage of a CVT is that, with an infinite number of gear ratios between a fixed top and bottom "gear", the engine is always running at the most efficient speed. At least that is what happens in economy tests, which is why car companies like them so much.
Enthusiasts are less sure. The dull whine as the car accelerates is no fun, and the fact that engine speed appears to have no relation to road speed is disconcerting. Some CVT gearboxes have virtual gears so that the car appears to be stepping through a gearbox, which seems a bit daft for something that is meant to have an infinite number of gears.
We would much prefer a dual clutch transmission to a CVT, but we are only drivers. The fact that CVT can give an advantage on an economy test (if not in real-world conditions) makes it mighty attractive to the powers that be.