It's EU wide and not just EU-based airlines.
From Jan 1, 2010, operators of aircraft on eligible flights have to monitor their fuel burn and thus their CO2 emissions on an annual basis.
Any aircraft over 5700kgs, qualifies with some exemptions for commercial transport aircraft who emit less than 10,000 tonnes of CO2 per annum and who carry out less than 242 flights during consecutive four month preiods. Non commercial operators below those figures must still comply but qualifiy for simplified calculation procedures. VFR flights don't count.
Eligible flights include any flights departing frm or terminating in the EU so all the overseas carriers flying into the Union must also put in place their Monitoring Plans.
At the end of each monitoring period (year), tallies must be verified by independent audit and thereafter begins a process of trading under the ETS or Emissions Trading Scheme.
It's a huge undertaking for the larger carriers and still involves significant work for even single aircraft operators. We await with interest to see how they deal with our own helicopter (yes, it's over 5700kgs MTOW) which very often departs VFR, arrives VFR but changes to IFR en route............