Twisted conjugacy is a generalisation of the well-known notion of conjugacy in groups: given two groups G and H, two homomorphisms φ,ψ: H → G and two elements g, g' in G, we say that g and g' are (φ,ψ)-twisted conjugate if there exists some h in H such that g = ψ(h)⁻¹ g' φ(h). In this seminar, I will discuss three search problems related to twisted conjugacy, and present the current progress towards constructing algorithms that solve these problems for (homomorphisms between) virtually polycyclic groups.