In graph theory, a cycle in a graph is hamiltonian if it visits every vertex of the graph. Determining hamiltonicity is difficult and one of Karp's famous 21 NP-complete problems. Applications of hamiltonicity range from combinatorial optimisation – in particular, the Travelling Salesman Problem – and operations research over coding theory, theoretical computer science, molecular chemistry, and fault-tolerance in networks. My project is closely linked with the study of hamiltonian cycles and related structures. I will address a series of interconnected conjectures and problems on longest cycles and spanning subgraphs, with a special emphasis on planar 3-connected graphs, i.e. the 1-skeleta of polyhedra.