Update: It seems like this deal has now ended.
American Airlines currently has very cheap fares on flights to Asia from the US. Depending on where you depart from, fares are pricing out at less than $400 roundtrip. For example, I was able to find many sub-$400 fares from Los Angeles to Hanoi, Ho Chi Minh City and Kuala Lumpur. Fares around $500 exist to Jakarta, Seoul and more.
There are a number of different options available and you can search in Google Flights and then click through to book on the American Airlines website. For more info on prices and available routes, see: this The Flight Deal post.