• Air France  from Paris Ch. de Gaulle (CDG) with 6 weekly flights,  ensures regular flights to Tahiti (PPT) via Los Angeles (Paris-Los Angeles = a 11h30 flight, stop in Los Angeles = 1h30, Los Angeles-Tahiti = 8h00, i.e. a total of 21 hours).
• Air Tahiti Nui  the Tahitian international airline (not to be confused with Air Tahiti,  the domestic airline that ensures inter-island flights only) offers regular flights on Airbus A340's from several cities: Paris (CDG) = 2 to 3 weekly flights, Los Angeles (LAX) = 3 weekly flights, Japan (Tokyo NRT or Osaka KIX) = 2 weekly flights, and Auckland (AKL) in New Zealand = 1 weekly flight. 
• Air New Zealand  provides 3 weekly flights between Auckland (AKL), Tahiti (PPT) and Los Angeles (LAX), with possible stopovers in Fiji and the Cook Islands. This airline also flies to Sydney (SYD) in Australia with a stop in  Auckland. For the british or europeans wishing to depart from London (LHR) rather than Paris, this airlines offers an alternative route with its flights London-Los Angeles-Tahiti-Auckland.
• Lan Chile  ensures flights twice a week from and to Chili (SCL), with a possible stopover in Easter Island.
• Hawaïan Airlines  offers two weekly flights to Honolulu (HNL), with connections to many destinations.
• Finally, Aircalin  ensures regular flights to New Caledonia and some Pacific Islands (Fiji, Wallis & Futuna, Noumea (NOU),  Vanuatu and New Zealand. It also flies to Sydney (SYD) Australia after a stop in Noumea.

Our advice: If you come from Europe and the length of the trip scares you (21 hours), it is often  possible to spend a day or two in Los Angeles, to break it into two shorter trips. Inquire before your buy your tickets.

And also...

Inter island planes and boats.

Air fares from Paris: Expect to pay between 1,000 Euros (low season outside school holidays) and 1,600 Euros (high season, in July-August and Christmas) for a round trip ticket, or keep an eye on tour operators' specials during the low season (March to early June and also mid-September to mid-December). Tickets are sometimes auctioned on specialized websites. A tourist recently paid only 500 Euros for Paris-Papeete round trip in April!!!