Abstract: We give a brief introduction to geometric structures on 3-manifolds, and various outstanding conjectures, in particular Thurston's conjectures about covers of 3-manifolds. We will also discuss some other topological structures on 3-manifolds and attempt to relate them to their geometries. We will mention how this relates to my own work, and other areas of maths, such as minimal surfaces and Lubotzky's work on discrete groups.