Real growing networks such as the World Wide Web or personal connection based networks are characterized by a high degree of clustering, in addition to the small-world property and the absence of a characteristic scale. Appropriate modifications of the (Barabási-Albert) preferential attachment network growth capture all these aspects. We present a scaling theory to describe the behavior of the generalized models and the mean-field rate equation for clustering. This is solved for a specific case with the result C(k) approximately 1/k for the clustering of a node of degree k. This mean-field exponent agrees with simulations, and reproduces the clustering of many real networks.