In the closing years of this century, organized crime syndicates have a problem with the standard tactic of threatening to kill someone for not cooperating because it appears that every person on Earth is tracked and can't die without leaving vital information about their death behind. By a lucky coincidence, however, it turns out that time travel has been invented and while it quickly becomes illegal, it does offer a rather neat solution for the elimination of these undesireables: send 'em into the past, where bad things can happen without leaving those awkward traces. Better yet, have people poised and ready to kill them as soon as they pop out of the future, "loopers".
The main character is a Looper named Joe (Joseph Gordon-Levitt), who exists 30 years in the past (2044) who is supplied with specific details of when others will be sent back, where they'll be at the precise moment they appear, and what to do: kill them immediately upon arrival. Conveniently, each has silver bars attached to their bodies, meaning that once they're dead, the looper gets paid. Disposing of the body's easy too as Joe works in a rural area.
Like any other job with the mob, loopers get snuffed out too, for a variety of reasons. To retire a looper, the mobster simply sends the older self back through the time portal with a hood over their head: the younger self then kills the older self without realizing ("closing the loop"), then figures out what's happened and knows that they have thirty years to live, after which point their death has already happened ("will already happen"?). There's a huge paradox here, but let's skip it, along with the question of why sending someone back in time defeats the tracking system but that, say, encasing them in concrete doesn't.