The Universe is inhomogeneousÐand essentially fractalÐon the scale of galaxies and clusters of galaxies, but most cosmologists believe that on larger scales it becomes isotropic and homogeneous: this is the`cosmological principle'. This principle was ®rst adopted when observational cosmology was in its infancy, and was then little more than a conjecture. The data now available offer a quantitative picture of the gradual transition from small-scale fractal behaviour to large-scale homogeneity.A fractal is a distribution or shape that is not homogeneous (in general), but possesses the property that each part is a simulacrum of the whole. In other words it`looks' the same on all scales. Fractals abound in nature: for example, the coastline of a small peninsula drawn on paper could equally well depict a large continent; and fractals have long been studied as manifestations of scaling laws in solid-state physics 1,2 . The clustering of galaxies (see Fig. 1) lends itself to a fractal description 1±7 , because the clumpiness prevails over a wide range of scales. The big question for cosmology is whether the distribution of matter continues to be a simple fractal beyond the scale of clusters of galaxies: if it does, then the cosmological principle is invalid.Past attempts to determine the distribution of matter have confronted two important obstacles. Most of the mass in the Universe is dark; we can detect it only by its gravitational effect on objects that we can see, and it is still unclear how to relate the distributions of light and mass. In particular, we have not known how to match the clustering of galaxies with the cosmic microwave background (CMB) anisotropies, which tell us about the mass uctuations in the early Universe. In addition, little has been known about¯uctuations in the distribution of matter on scales intermediate between those of local galaxy surveys (( 100 h -1 Mpc, where h is the Hubble constant in units of 100 km s -1 Mpc -1 ) and scales probed by the Cosmic Background Explorer (COBE) satellite () 1,000 h -1 Mpc).Recent observations of radio galaxies and the X-ray background (XRB, which probably arises from distant active galactic nuclei), at median redshift Å z < 1, can now effectively probe these intermediate scales.Moreover, experiments to measure¯uctuations in the microwave background on smaller angular scales than those probed by COBE are also helping to bridge the gap. Future surveys of more than one million galaxies will probe to a median redshift Å z < 0:1 (which roughly corresponds to a co-moving distance of ,300 h -1 Mpc). Current data, however, already strongly constrain any non-uniformities in the galaxy distribution (as well as the overall mass distribution) on scales ) 300 h -1 Mpc.In the language of fractals (see Box 1), fractal dimensions are used to characterize the degree of clustering: these generalize our intuitive concepts of dimension and can take any positive value less than or equal to 3. On scales below ,10 h -1 Mpc, galaxies are distributed with the correlation dimension...