The class of Artin groups is easy to define, via presentations,
which have the form
\(
\langle x_1,x_2,\cdots,x_n \mid \overbrace{x_ix_jx_i\cdots}^{ m_{ij}}= \overbrace{x_jx_ix_j \cdots}^{m_{ij}}, i\neq j \in \{1,2,\ldots,n\}\rangle
,\, m_{ij} \in {\mathbb{N}} \cup \{ \infty\}, m_{ij} \geq 2.
\)
But it contains a variety of groups with apparently quite different properties.
For the class as a whole, many problems remain open, including the word problem;
this is in contrast to the situation for Coxeter groups, which arise as quotients of Artin groups.
I'll discuss what is known about rewrite systems for Artin groups, and evidence
for the possibility of a general approach to rewriting in these groups.
I'll give some general background,
starting at work of Artin, then Garside, Deligne, Brieskorn-Saito,
then move on, via Appel-Schupp, to
very recent work, by myself and Derek Holt (and sometimes Laura Ciobanu), by Eddy Godelle and
Patrick Dehornoy, also by Blasco, Huang-Osajda.