Cheeger-type inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis, in particular in the context of expander theory. The natural problem to extend such inequalities to simplicial complexes and their higher order Eckmann Laplacians has been open for a long time. Before proving any inequality, however, one needs to identify the right Cheeger-type constant for which such an inequality can hold. Here, we solve this problem. Our solution involves and combines constructions from simplicial topology, signed graphs, Gromov filling radii and an interpolation between the standard 2-Laplacians and the analytically more difficult 1-Laplacians, for which, however, the inequalities become equalities. It is then natural to develop a general theory for $p$-Laplacians on simplicial complexes and investigate the related Cheeger-type inequalities.