(Joint work with Maria Cumplido and Ruben Blasco) Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. In particular, it is unknown whether the word problem is solvable for all Artin groups. I will discuss a new algorithm for solving the word problem in 3-free Artin groups. This is based on work by Holt and Rees for large type and sufficiently large type groups (2012 and 2013). Our work significantly broadens the class of Artin groups to which this result applies because it allows for groups that have commuting generators. This algorithm gives an explicit way to reduce a word to a geodesic word without ever increasing the length of the word.