Coxeter groups arise in many areas of mathematics including topology, algebraic geometry, representation theory, and combinatorics. They are much studied and generally well-understood. Closely related to Coxeter groups are a class of groups known as Artin groups. The best known examples of these groups are the braid groups. While easy to define, Artin groups are much more mysterious than Coxeter groups with many key properties unknown and basic questions unanswered. In this talk, I will introduce Artin groups, discuss some of the open problems, and survey some recent work, using both combinatorial and geometric methods, to approach these problems.