Trains with rubber tyres provide higher traction and a smoother ride. They are found in a good bunch of monorails and subways around the world.
France has been the pioneer, starting with experimental rubber-tyred railway trains as early as 1929. In 1946, tyre manufacturer Michelin has patented a steel-belt rubber tyre. It was used from 1951 in experiments on the Paris Metro, leading to regular service on metro line 1 from 1956 till today.
The engineers in Paris came to the conclusion that the rubber-tyred system had three times the adhesion of steel-on-steel systems and could significally reduce vibrations and noise for passengers and neighboring residents alike. The bogies in Paris' metro trains do not only have rubber-tyred wheels but also backup steel wheels and rubber-tyred horizontal guiding wheels. Trains run on a concrete surface with auxiliary steel rails. Today, five metro lines run on rubber. The reason that not the entire Paris metro network has been converted to rubber-tyred operation was the cost of retrofitting of the horizontal guidance infrastructure.
Following the Parisian success story, the rubber tyre has been adopted by several metro systems around the world. Here's a list of cities with rubber-tyred urban transit lines. People movers listed only if part of urban transit.