Many of the optimization problems of interest to humanity are NP-hard, and most computer scientists believe that they have no efficient algorithms. Focusing on approximation rather than exact optimization might extend the reach of algorithmic technique into the intractable. However, for many NP-hard problems approximating them better than a problem-dependent threshold turns out to be NP-hard as well. Proving so rigorously is a difficult task, which -- by a leap of thought -- leads to fundamental questions about the nature of proofs and their verification.

In this talk I'll discuss the phenomenon of sharp thresholds in approximability, namely how many approximation problems transform instantly from efficiently solvable to exponentially hard as one focuses on a better approximation (joint work with Ran Raz). I'll discuss two prover games and a new, incredibly simple, method ("fortification") for analyzing their parallel repetition. Finally, I'll discuss a recent candidate hard instance for unique games, which might lead to progress on the Unique Games Conjecture - one of the biggest open problems in approximability (joint work with Subhash Khot).