The focus of this survey is the question of *quantified derandomization*, which was introduced by Goldreich and Wigderson (2014): 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?

This question opens the door to studying natural relaxed versions of the derandomization problem, and allows us to construct algorithms that are more efficient than in the general case as well as to make gradual progress towards solving the general case. In the survey I describe the body of knowledge accumulated since the question's introduction, focusing on the following directions and results:

1. *Hardness vs ``quantified'' randomness:* Assuming sufficiently strong circuit lower bounds, we can derandomize probabilistic algorithms that err extremely rarely while incurring essentially no time overhead.

2. For general probabilistic polynomial-time algorithms, *improving on the brute-force algorithm for quantified derandomization implies breakthrough circuit lower bounds*, and this statement holds for any given probability of error.

3. Unconditional *algorithms for quantified derandomization of low-depth circuits and formulas*, as well as *near-matching reductions* of the general derandomization problem to quantified derandomization for such models.

4. *Arithmetic quantified derandomization*, and in particular constructions of hitting-set generators for polynomials that vanish extremely rarely.

5. *Limitations of certain black-box techniques* in quantified derandomization, as well as a tight connection between black-box quantified derandomization and the classic notion of *pseudoentropy*.

Most of the results in the survey are from known works, but several results are either new or are strengthenings of known results. The survey also offers a host of concrete challenges and open questions surrounding quantified derandomization.