A Flyway is the name that has been given to the broad corridors where migratory birds fly from the northern to the southern hemisphere and back again.  There are eight flyways that are identified around the world: - The Pacific Americas Flyway, the Mississippi Americas Flyway, The Atlantic Americas Flyway, The East Atlantic Flyway, The Black Sea/Mediterranean Flyway, The East Africa West Asia Flyway, The Central Asia Flyway, and the East Asia Australasian Flyway. 
There is some over-lap with flyways.  Although most birds will follow the same routes down the flyway, there are always some that take a slightly different route and end up in an entirely different place.   This is what makes finding a 'rarity' so exciting. 
Shorebirds rest and feed at different wetlands along their routes and it is at such places that banded and flagged birds are reported.  Over time this has built up knowledge of the routes different birds take and the places where they rest.  Recently, scientists have been using some very hi-tech solutions to get details of specific species which use the flyway.  
In 2007 scientists in New Zealand attached satellite transmitters to Bar-tailed Godwits (Limosa lapponica baueri).  One bird, E7, was tracked on her full migration both north and south.  Her migration re-wrote the books as until then there were no flyways known to cross the Pacific Ocean directly.  You can read about it here:
The next year satellite transmitters were attached to Bar-tailed Godwits in NW Australia.  These birds are from the sub-race menzbieri and their route is slightly different.  You can read about it here and see the map of their tracks.  In 2009 geo-locators were attached to Ruddy Turnstones in Victoria, Australia, and their tracking was just as interesting and unexpected.  One of the birds travelled back across the Pacific with one stop at the Gilbert Islands.  You can read about it here:
With the results of the Godwit tracking and also that of the Ruddy Turnstone, the East-Asian Australasian flyway has been changed to include the routes of these two species.  However, I notice on a recent flyway map that it is suggested that a new flyway be recorded and called the West Pacific Flyway.