  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.
