The algebraic combinatorial Garside theory is a method to transport information from a Coxeter group to its related Artin group. It has been succesfully applied to Coxeter and Artin groups of spherical and of affine type, yielding for instance a solution of the word problem in the related Artin group. In the study of the Artin groups of affine type first Digne relaxed the condition that the set of atoms of a Garside structure has to be finite, then McCammond-Sulway and Paolini-Salvetti overcame the restriction that the set of atoms forms with respect to the given order relation a lattice. In the talk I will propose a systematic study of the possible set of atoms in a Coxeter group. If time is left I will also discuss extended Weyl groups.