Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, a submonoid of the Artin group, then some aspects of Garside theory still apply in the infinite-type case. I will talk about joint work with Rachael Boyd and Rose Morris-Wright on geometric relations between Artin monoids and Artin groups.