The discovery of topological order in the late 80s brought about a paradigm shift in our understanding of phases of matter. Models of topological order in two spatial dimensions famously host a variety of remarkable phenomena such as excitations with anyonic statistics. Anyons are point-like particles with exotic statistics, i.e. neither bosonic nor fermionic, making them prime candidates for quantum memories and quantum computing platforms. Exchange statistics of point-like particles may be studied in terms of the braids that form their worldlines in 2+1 (spacetime) dimensions, so that anyonic statistics is governed by representations of the so-called braid group. Crucially, since such braids can always be disentangled in 3+1 dimensions, anyons can only exist in two-dimensional systems. However, three-dimensional topological models are known to yield higher-dimensional excitations, which also have unconventional statistics, and as such behave like extended anyon-like objects. The goal of my research project is to boost our understanding of higher-dimensional topological models---which are far less understood than their two-dimensional counterparts---from the viewpoint of these excitations. More specifically, I intend to devise tools to study the extended excitations, address the notion of duality in three-dimensional models, and develop a tensor network framework for higher-dimensional topological phases.