Large-scale structure in the distribution of galaxies is thought to have evolved through gravitational instabilities from small density fluctuations in the (largely homogeneous) early Universe. This structure of galaxies consists of rich and poor clusters, connected by filaments and sheets, with regions largely devoid of galaxies (voids) in between(1). Numerical simulations of the growth of initial density fluctuations through a nonlinear regime, motivated by the likely physics of the early Universe, also show a network of filaments and voids(2,3,18), but the origin of this picture of filaments as the dominant structure was not well understood. Here we show that the ’web’ of filaments that defines the final state in these simulations is present in the initial density fluctuations; the pattern of the web is defined largely by the rare density peaks in the initial fluctuations, with the subsequent nonlinear evolution of the structure bringing the filamentary net work into sharper relief. Applying these results to the observed galaxy distribution, we suggest that ’superclusters’ are filamentary cluster-cluster bridges, and we predict that the most pronounced filaments will be found between clusters of galaxies that are aligned,vith each other and close together.