Hyperbolic geometry is the non-Euclidean geometry discovered by János Bolyai, N. I. Lobachevsky, and K. F. Gauss about 200 years ago. Work of George Daniel Mostow, William Thurston, and Gregory Perelman shows that hyperbolic 3-manifolds are the most important class of three-dimensional (3-D) manifolds. Therefore, it is of great importance to come to a deep understanding of hyperbolic 3-manifolds. The first part of this article begins with basic ideas, but ultimately focuses on the most natural tool for analyzing a hyperbolic 3-manifold: its volume. The second part of the article investigates the geometric rigidity of hyperbolic 3-manifolds.