The Odyssey was one of Western civilization's first masterpieces of literature, composed by Homer, about the ten year voyage of Odysseus, a king who fought in the Trojan War. Suffice to say he offended a Greek god and was forced to take the long way home.
Borrowing from this original meaning, an odyssey is any long foreign trip.
Example: An upcoming trip to Asia by President Obama rumored to be lavishly expensive, worthy of Roman Emperors or Queen Cleopatra of Egypt, was referred to by the Drudge Report in this way:
The true expense of the voyage, er, trip, is unknown.