The short answer is that a company called Premiere Exhibitions in Atlanta, Georgia currently owns the salvage rights to the Titanic shipwreck.
The long answer is, it’s a bit tricky.

In 1985, the Titanic shipwreck was discovered in 12,000ft deep water of the North Atlantic Ocean some 350 nautical miles off the coast of Newfoundland, Canada. The wreck was “co-salvaged” by Titanic Ventures Incorporated and Ifremer. By 1987, they had liberated over 1,800 artifacts from the Titanic shipwreck.

According to the NOAA, “the primary purpose of the Maritime Law of Salvage is to prevent looting of recent marine casualties.”

In 1994, Titanic Ventures changed their name to RMS Titanic Incorporated (which is a subsidiary of Premier Exhibitions Incorporated) and obtained exclusive salvage rights. Since then, they’ve completed 8 research exhibitions and salvaged 5,300 artifacts.
BONUS:
TAKE A $60,000 SUBMARINE DOWN TO THE TITANIC SHIPWRECK!!!!!
You will ride in a nickel-steel Russian MIR sub and the entire journey will be an unforgettable 11-12 hour experience. It takes 2.5 hours to get there and you will spend 3-4 hours down there. Incredible.