The term "MO-Hard" has entered the mathematical vocabulary, describing interesting questions that were asked on MathOverflow but haven't been answered in spite of efforts by many community members. Each one of these is a MathOverflow success story! Open problems are precious commodities that have driven mathematical research for centuries. While these are not necessarily at the same level as the Riemann Hypothesis and the Jacobian Conjecture, we should celebrate them and promote them!
While the unanswered tab is a good way to find such gems, questions that have interesting partial answers do not appear there. In the same spirit as Best of MathOverflow, let's collect MO-Hard questions here. Questions that still appear in the unanswered list are welcome since they might disappear any day. Comments, anecdotes and other remarks are also encouraged.