Artin groups form a large class of infinite groups. They contain groups such as free abelian groups, free groups, braid groups, and many others. Despite an increasing interest in the topic, these groups remain quite mysterious in general, in the sense that many classical questions in geometric group theory are still unanswered for them. The first half of the talk will serve as an introduction to Artin groups (and Coxeter groups, which are their cousins) and some standard tools in the theory. In the second half, I will talk specifically about the isomophism problem, and I will give a solution to it for certains families of Artin groups.