The Cheeger inequality for graphs links connectivity properties of a graph with the spectrum of its Laplacian. We will start with a brief introduction to the ideas behind spectral graph theory, focusing on the information available from the spectrum of the graph Laplacian. We will then sketch a proof of the Cheeger inequality, talk about its significance. If time permits, we will end with a discussion of ways to generalize the Cheeger inequality to structures like vector bundles and simplicial complexes.