Beijing's airport express opened before the 2008 Olympics and provides 25 minute service directly from both terminals of Beijing's airport to two stations in the heart of the city, Dongzhimen and Sanyuanqiao.
Seoul's AREX (Airport Rail Express) opened in 2010. Previously, Incheon Airport's long distance from Seoul made transit difficult. But the new line cuts time to around 50 minutes, and the line provides direct express service to Seoul Station in addition to the "all-stop train" serving several stops along the way including Gimpo Airport. Recently, Korea's KTX High Speed Rail opened service directly from Incheon Airport to the rest of Korea, making Incheon's rail transit connections some of the best in the world.
The clunky American answer
The proposed trains to LaGuardia in New York, LAX in Los Angeles, and Oakland all demonstrate the absolute WRONG way to go about connecting airports with public transit. The transfers, lack of direct access to downtown areas, and difficulty moving luggage through these transfers all work to discourage use of airport rail. In many cases, taking the airport rail will be more trouble than its worth.
Transfers: Instead of routing a metro line directly into the airport terminal area, the systems will require passengers to board a light rail, then transfer to another line some distance away from the airport. In New York's case, the proposed Air Train to LaGuardia will dump passengers at Flushing Meadows Station on Line 7, requiring a transfer and then a long shlep to Manhattan via the Subway or the LIRR. New York's JFK and Newark Airports do have connections, but they are similar: an airport train connects with Jamaica Station for JFK, and Newark's Rail Station in Newark. Because business travelers usually need to get to central city districts quickly, the complicated transfers discourage them from taking advantage of the train. Tourists must navigate the complicated and unintuitive path from the Airport Train to commuter rail or subways.
Cost vs Benefit
In all the American examples, the high cost of building direct rail into the airport is usually the most important reason cited by (politicians/activists/insert interest group here) to claim that people-movers or trams are the most cost effective way to connect airports to transit systems. But when you realize that the inconvenience discourages many passengers from taking rail to the airport, the cost savings in initial construction has to be considered against the lost revenue from lower ridership.
Los Angeles: Failure in the Making