Recent studies have revealed characteristic general features in the topology of real-world networks. We investigate the universality of mechanisms that result in the power-law behaviour of many real-world networks, paying particular attention to the Barabasi-Albert process of preferential attachment as the most successful. We introduce a variation on this theme where at each time step either a new vertex and edge is added to the network or a new edge is created between two existing vertices. This process retains a power-law degree distribution, while other variations destroy it. We also introduce alternative models which favour connections to vertices with high degree but by a different mechanism and find that one of the models displays behaviour that is compatible with a power-law degree distribution.