The focus of this monograph is the question of quantified derandomization. Does derandomization of probabilistic algorithms become easier if we only want to derandomize algorithms that err with extremely small probability? How small does this probability need to be in order for the problem's complexity to be affected? In this comprehensive survey of the topic, the author describes the body of knowledge accumulated since the question was first raised in 2014, focusing on the following directions: 1) Hardness vs "quantified" randomness; 2) Improving on the brute-force algorithm for...