  This thesis deals with the conjugacy problem in groups and its twisted
variants. We analyze recent results by Bogopolski, Martino, Maslakova and
Ventura on the twisted conjugacy problem in free groups and its implication for
the conjugacy problem in free-by-cyclic groups and some further group
extensions. We also consider the doubly-twisted conjugacy problem in free
groups. Staecker has developed an algorithm for deciding doubly-twisted
conjugacy relations in the case where the involved homomorphisms satisfy a
certain remnant inequality. We show how a similar condition affects the
equalizer subgroup and raise new questions regarding this subgroup. As an
application we discuss the Shpilrain-Ushakov authentication scheme based on the
doubly-twisted conjugacy search problem in matrix semigroups over truncated
polynomials over finite fields. Part of this thesis is devoted to the
implementation and testing of some of the previously mentioned concepts in the
GAP programming language.
