"This weapon is your life," Obi-Wan Kenobi once said to Anakin Skywalker in Star Wars: Attack of the Clones, and this couldn't be more true. The process of making a lightsaber is personal, and this weapon often reflects their Jedi in some way, shape, or form. Despite the importance of a lightsaber, Luke Skywalker's lightsaber is only present in the sequel trilogy during a flashback of Luke attempting to kill Ben Solo in The Last Jedi.
Outside of this memory, the green lightsaber is nowhere to be seen. Instead, Luke and Rey use Anakin's former, blue lightsaber for most of the sequel trilogy, and at the end of The Rise of Skywalker, Rey ceremoniously buries this lightsaber with Leia's as a farewell to her two masters. While Anakin's lightsaber is meant to stand in for Luke's at this moment, there is still no sign of Luke's true lightsaber after the tragic events of the Jedi Academy.
Several theories have popped up since Luke was brought into this new trilogy, and after The Rise of Skywalker failed to give any closure about what happened to this iconic weapon, only more have come forward. One of the more legitimate theories ties back to The Last Jedi novelization by Jason Fry, which goes more in-depth about Luke's time on Ahch-To.
"[Alcida-Auka] directed another daughter to take his weapon, his star compass and his strange other gear to the repository, where it would join other items gathered over the generations."
From the novelization, it's clear that when Luke arrived at Ahch-To years ago, the Lanai confiscated most of his belongings, locking them away for the indefinite future. This includes a nondescript weapon, which could've been his green lightsaber. Given the fact that the last known use of this lightsaber was an attempt on his nephew's life, it wouldn't be shocking to learn that Luke willingly let go of this weapon since it's now tied to this traumatic event. It's even in character for this trilogy's Luke to do so since he's actively turned his back on the Jedi way and carelessly threw away his father's lightsaber when presented with it.
While this idea makes sense, the novelization provides no description of what the weapon is, which leaves room for fans to doubt if it even was Luke's lightsaber at all. Given this vague explanation, fan theories continue to garner support, and one Reddit theory, from RDruner870, has done particularly well.
According to this post, Luke's lightsaber is hidden in R2D2's storage compartment. The droid was present at the Jedi Academy during Kylo Ren's attack, and as the place burns to the ground, he is beside Luke. It's also established that Luke had his lightsaber prior to this attack, so it's safe to assume he still has it as everything burns around him.
Instead of bringing his lightsaber to Ahch-To, Luke could've hidden it with R2d2, like he hid the remaining piece of the map to his secretive location. While Luke was in hiding, he did provide clues for the Rebels to find him in case of the worst, so he could've also left behind his lightsaber as another precaution.
Furthermore, it's plausible that R2D2 has the lightsaber given the events of Return of the Jedi. When Luke and company go to save Han Solo, R2D2 has a compartment that fits the lightsaber perfectly, and the droid can eject the weapon at a moment's notice. Luke could've trusted R2D2 with the lightsaber after the events of the Jedi Academy, hoping the droid would release it if needed in the future.
This theory also believed that The Rise of Skywalker would see R2D2 eject the lightsaber, like he did in Return of the Jedi; however, this time Rey would be the one to catch it since Luke died in the prior film. As seen in the finalized movie, this doesn't happen, but with the lightsaber missing in action on screen, it's still possible that R2D2 is holding onto the weapon since his master is no longer there to give him an order.
Another theory is that Kylo took Luke's lightsaber during the attack on the Jedi Academy, which YouTuber Star Wars Analyst goes in-depth about. A Jedi's lightsaber is made with a colorless kyber crystal, which takes on the color corresponding to the force user's spiritual alignment; however, kyber crystals can't naturally become red.
As established in the 2017 Darth Vader comics, the way a Sith gets a red lightsaber is by pouring their rage and pain into it, which is known as bleeding. The other key part to constructing a red lightsaber is that the crystal being used must be taken from someone else. While Kylo could've stolen any of the students' crystals, stealing the one from Luke's lightsaber would make the most sense given the amount of rage Kylo feels towards his uncle.
Based on this theory, this would mean Luke's lightsaber no longer exists. This explains why the only time it is seen on screen is moments before Kylo defends himself and blasts Luke back with a devastating force push. This push could have also cracked Luke's crystal, which further strengthens this theory since The Force Awakens' Visual Dictionary confirms that Kylo's unstable lightsaber design is due to a cracked kyber crystal.
Without getting confirmation on screen about the whereabouts of Luke's lightsaber, speculations will continue. While The Last Jedi novelization confirms some form of weapon was confiscated from the Jedi on Ahch-To, it's not clear enough to provide a definitive answer for fans. The green lightsaber could be with the Lanai, but it could also be locked inside R2D2, or it is at the bottom of the ocean with the rest of Kylo's crossguard lightsaber.