For graphs, the notion of expansion is a highly useful concept that has found applications in various areas, especially in combinatorics and theoretical computer science. In recent years, the success of this concept has inspired the search for a corresponding notion in higher dimensions. Graph expansion can be expressed spectrally (via the spectrum of the Laplacian or the adjacency matrix) or combinatorially, e.g., as the edge expansion ratio. The close connection of these two notions is expressed, e.g., by the discrete Cheeger inequality. This talk addresses generalizations of expansion properties to finite simplicial complexes of higher dimension. Whereas for spectral expansion there is an obvious candidate - higher dimensional Laplacians were introduced already in the 1940s by Eckmann - the generalization of edge expansion to simplicial complexes is not straightforward. In this talk, we will focus on a topologically motivated notion analogous to edge expansion, based on Z_2-cohomology, that was recently introduced by Gromov and independently by Linial, Meshulam and Wallach.