The most basic Riemannian manifolds are those admitting a complete metric of constant curvature. The classification of closed manifolds with metrics of positive and zero curvature is has been relatively well-understood for quite a long time. Constant curvature -1 manifolds, hyperbolic manifolds, remain quite a bit more mysterious, particularly in high dimensions. I will give a (biased) narrative regarding what we know, including a number of exciting recent results with connections to dynamics and geometric group theory, and look forward to some problems I hope to see solved in the coming years.