Future 5G services are characterised by unprecedented need for high rate, ubiquitous availability, ultralow latency, and high reliability. The fragmented network view that is widespread in current networks will not stand the challenge posed by next generations of users. A new vision is required, and this paper provides an insight on how network convergence and application-centric approaches will play a leading role toward enabling the 5G vision. This paper, after expressing the view on the need for an end-to-end approach to network design, brings the reader into a journey on the expected 5G network requirements and outlines some of the work currently carried out by main standardisation bodies. It then proposes the use of the concept of network convergence for providing the overall architectural framework to bring together all the different technologies within a unifying and coherent network ecosystem. The novel interpretation of multidimensional convergence we introduce leads us to the exploration of aspects of node consolidation and converged network architectures, delving into details of optical-wireless integration and future convergence of optical data centre and access-metro networks. We then discuss how ownership models enabling network sharing will be instrumental in realising the 5G vision. This paper concludes with final remarks on the role SDN will play in 5G and on the need for new business models that reflect the application-centric view of the network. Finally, we provide some insight on growing research areas in 5G networking.