  We survey the current state of affairs in the study of thresholds and sharp
thresholds in random structures on the occasion of the recent proof of the
Kahn--Kalai Conjecture by Park and Pham and the fairly recent proof of the
satisfiability conjecture for large $k$ by Ding, Sly, and Sun. Random discrete
structures appear as fundamental objects of study in many scientific and
mathematical fields including statistical physics, combinatorics, algorithms
and complexity, social choice theory, coding theory, and statistics. While the
models and properties of interest in these fields vary widely, much progress
has been made through the development of general tools applicable to large
families of models and properties all at once. Historically these tools
originated to solve or make progress on specific, difficult conjectures in the
areas mentioned above. We will survey recent progress on some of these hard
problems and describe some challenges for the future.
