It is possible to define limiting objects that can approximate dense networks, as the number of vertices tend to infinity, in much the same way that the normal distribution approximates that of a sum of independent random variables. In this way, properties of dense graphs can be approximated by those of a suitably chosen limit. A network on n vertices is associated, via its adjacency matrix, with a symmetric function on the unit square that takes values of either 0 or 1, constant over squares of side 1/n, unique up to permutation of the vertex labels. This representation of a network belongs to the larger space of equivalence classes of symmetric, measurable functions on the unit square taking values in the unit interval, with two functions equivalent if one can be obtained from the other by a measure-preserving transformation of the axes; the equivalence classes are called graphons. A metric is defined on the space of graphons with the property that a sequence of ever larger dense networks converges, with respect to this metric, to a limiting graphon W, if, for each k, the distribution, over the set of possible subgraphs of size k, of a randomly chosen induced subgraph of size k in the network converges to a corresponding distribution derived from W.