Buster's Trip

it is called busters around the world trip
In the Season 2 episode "Arthur's Faraway Friend," Buster left Elwood City to travel with his dad. He did not return until the beginning of Season 3.

Buster Leaves Town
Because Buster's parents are divorced, he never really gets to spend a lot of time with his father. When Buster's dad, a pilot, invited Buster to travel with him for a couple of months, Buster initially did not want to go, as he would miss all his friends, but eventually decided he did want to go as it meant that he would get to spend some quality time with his dad, as well as see the world. When Buster received the invitation to go on the trip, he and Arthur were preparing to write an exciting adventure book, and with Buster out of town it would mean that they wouldn't be able to work on the story. Fortunately, Buster sent Arthur a list of the places he was going so they could get letters to each other, and were able to send each other the chapters. Arthur decided to use Buster's traveling to their advantage, as they could feature all the places Buster was going in the story.

At first Arthur was very sad that Buster was leaving, but he eventually was okay with it. Nonetheless, he eagerly looked forward to when Buster returned home.

While Buster was gone
A number of events happened in Elwood City while Buster was away. Arthur became a champion of the trivia game "No Guessing" (though he actually cheated), Muffy hosted a party at Waterworld and a carnival in her backyard and later launched a line of cookies with the help of friends. Arthur, The Brain, and Binky found a mysterious key, and a scary new crossing guard was hired at Lakewood Elementary.

Arthur kept Buster informed of all the goings-on in Elwood City through letters. He even thanked Buster for his advice in order to overcome his cheating problem, even though Buster never actually gave him advice.

Buster returns
Eventually, Buster's mom called Arthur to inform him that Buster would be coming home later that day. Arthur was very excited at first, but when D.W. told Arthur that Buster might not want to be Arthur's friend anymore after his exciting world travels, Arthur grew very worried. Fortunately, Buster had not changed at all during the time he was gone, and was eagerly looking forward to playing a game of checkers with Arthur, just like old times.

However, Buster began to feel left out once he learned about all the things that happened while he was away. Arthur and his friends decided to throw a party for him and watch his travel slides, and Buster was glad they all still wanted to be his friend.

Trivia

 * Apparently, when Buster returned to town, he unknowingly brought Art Garfunkel with him. A possible explanation is that Buster's dad was Garfunkel's pilot while he performed concerts.
 * Buster would later go on another trip with his dad in Postcards from Buster, though this time Arthur and his family went with him. This later inspired the spin-off series Postcards from Buster.
 * In the in the song The Ballad of Buster Baxter, there is a line "We thought he left the show." This could possibly be in response to worried viewers who had written to the show wanting to know if Buster was ever going to come back.