Search and witness problems in group theory

Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. On the other hand, witness problems are: given a property P and an object O with the property P, find a proof of the fact that O indeed has the property P. On the third hand(?!), search problems are of the following nature: given a property P and an object O with the property P, find something “material” establishing the property P; for example, given two conjugate elements of a group, find a conjugator. In this survey our focus is on various search problems in group theory, including the word search problem, the subgroup membership search problem, the conjugacy search problem, and others. To Alfred Lvovich Shmelkin with deepest appreciation