Cover-free families of sets, their generalizations and applications

We start from the following question:
What is the maximal number of subsets of a given finite set such that no one subset is covered by t others?
We present the history of this problem which was discovered under different names in group testing, coding theory and combinatorics. We consider variations of the problem, in particular, Renyi-Ulam search with a lie. Then we embed the problem into more general question:
How to find unknown subset of a finite set?