Expansion properties of graphs and, in particular, bounded degree expander graphs, are a central research topic in combinatorics and theoretical computer science, with many applications and connections to other areas of mathematics. In recent years, there has been a concerted effort to generalize and extend this rich and fruitful theory to higher dimensions. In this thesis, some of these recent developments and results are surveyed. Several notions of higher-dimensional expansion, in particular coboundary expansion (introduced in the work of LinialMeshulam and Gromov) on the one hand and spectral expansion formulated in terms of the eigenvalues of higher-dimensional Laplacians (going back to the work of Eckmann and Garland) on the other hand, are described and connections and differences between these are discussed. Moreover, several applications of higher-dimensional expansion, e.g., Gromovs topological overlap theorem are presented.