In this talk, I will explain the definition of {\it quasi--Coxeter algebras}, which are to generalised braid groups what Drinfeld's quasi--triangular, quasi--Hopf algebras are to Artin's braid groups. Quasi--Coxeter algebras are one of the main tools required to prove that the monodromy of the Casimir connection \cite {MTL} is described by Lusztig's quantum Weyl group operators \cite{TL}. I will motivate their definition by using De Concini and Procesi's compactifications of hyperplane complements which yields, in the case of the Coxeter arrangement of type $A_{n-1}$, the moduli space $\overline{\mathcal M}_{0,n+1}$ of stable, $n+1$--marked curves of genus zero. Embedded in these compactifications are some remarkable polytopes which generalise Stasheff's associahedra. Time permitting, I will also sketch the deformation theory of quasi--Coxeter algebras and describe the related controlling complex. \begin{thebibliography}{ZZZ} %===================== \bibitem[MTL]{MTL} J. J. Millson, V. Toledano Laredo, {\it Casimir operators and monodromy representations of generalised braid groups}, to appear in Transform. Groups {\bf 10} (2005), {\sf math.QA/0305062}. \bibitem[TL]{TL} V. Toledano Laredo, {\it Quasi--Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups}, in preparation.