James Mangold's Logan follows Hugh Jackman's eponymous Logan (also known as Wolverine), who is a dying man. His body is failing him; the adamantium in his body is slowly poisoning him, preventing him from using his powers to their full extent and slowly killing him.
Logan spends much of his time driving people around in his fancy car for money. He does this for two reasons: 1) to earn enough money to buy a house boat and 2) to take care of Professor X and all the medication that he needs to prevent him from causing a tremendous amount of damage with his seizures again.
One day, Logan and Professor X come into contact with a young mutant named Laura, whose powers are eerily similar to Logan's. The woman who Laura is with implores Logan to help her bring Laura to a place called Eden - a safe place, far away from the clutches of Alkali-Transigen and its evil minions (including Donald Pierce and his Reavers).
After some trepidation, Logan agrees to help bring Laura to Eden. He eventually succeeds in doing so, but not without some collateral damage: Professor X first dies and then Logan dies himself.