Currently, the flying sub only exists as a set of design objectives issued by DARPA (the Defense Advanced Research Projects Agency). They want a craft that can arrive at a target via air, hang out for a while as surface boat, and disappear stealthily beneath the waves. According to the official request for proposals:

"By combining the beneficial characteristics and operating modes of each platform, DARPA hopes to develop a craft that will significantly enhance the United States tactical advantage in coastal insertion missions."

The proposed craft would hold eight soldiers plus all their gear and could support them as a floating surface craft for 72 hours. It would have a 1,000 mile aerial range and a 12 mile submerged range. It would more accurately be called a submersible aircraft, as an aircraft design can be pressurized and submerged far more easily than a heavy submarine could be made to fly.

Oddly enough, Military.com points out that this idea has been floating around for decades, but no one's ever even finished a prototype. Here's an artist's rendering of a design from the 1960s: Images by: DARPA/AP.

Flying Submarine or Submerging Seaplane? [Military.com]