Contact geometry, in dimension three, lies at the intersection of many fields: symplectic geometry, 3- and 4-dimensional topology, CR-geometry, dynamics, etc. Contact structures come in two flavors: tight contact structures which tend to reflect the ambient geometry of the 3-manifold, and overtwisted contact structures which are flabby (homotopic in nature) and are relatively well-understood. In this talk, I will explain the dichotomy of 3-manifolds that carry finitely many tight contact structures vs. those that carry infinitely many. This is joint work with Vincent Colin and Emmanuel Giroux.