My goal is to advertise the richness of Garside structures among groups of geometric nature. I will first present some nonpositive curvature aspects of Garside groups. I will then show that affine buildings give rise to exotic Garside groups. I will explain a simple way to produce, starting from a certain group G, a Garside structure on the direct product G x Z. This applies to surface groups, to some systolic groups, and to some Artin groups. It applies notably to some Artin groups whose Dynkin diagram is a circle, yielding many strong consequences for them. This is a joint work with Jingyin Huang.