The Hawking- Penrose singularity theorems tell us that cosmological solutions of Einstein's equations generically contain a singularity. But these theorems tell us little about what happens near such a singularity. Do the gravitational fields necessarily grow without bound? Can causality break down? What about the Cosmic Censor? Work done during the past ten years--both analytical and numerical--has gotten us a lot closer to answers to these questions. We survey this work, discussing both the mathematical ideas and the physical implications. We also discuss the likely direction of future studies.