During his years as a Jedi, Obi-Wan Kenobi fought with three lightsabers. His first had a blue plasma blade and a hilt similar to that of his master, Qui-Gon Jinn. Obi-Wan used this weapon until Darth Maul kicked it into a deep shaft on Naboo. He then constructed a new, blue-bladed lightsaber with a similar design. After attaining the rank of Jedi Master, Obi-Wan built a third saber, also with a blue plasma blade. He used this saber until he sacrificed his life aboard the Death Star, allowing himself to be struck down by Darth Vader so Luke Skywalker could escape.
What happened to Obi-Wan’s lightsaber after A New Hope? In Legends, Vader held onto it after Kenobi’s death and stored it in his fortress, where it remained for years. It was eventually recovered by Anakin Solo, one of the children of Han and Leia. Luke Skywalker held onto his old mentor’s lightsaber after that. We don’t have a canon answer for this question, but for now I assume it was destroyed along with the Death Star. I like the idea of Vader again keeping it in his fortress, but I also think it’s unlikely he would bring the lightsaber with him onto his starfighter in his pocket or something.