In this article, we address the prospects and key enabling technologies for highly efficient and accurate device positioning and tracking in 5G radio access networks. Building on the premises of ultra-dense networks as well as on the adoption of multicarrier waveforms and antenna arrays in the access nodes (ANs), we first formulate extended Kalman filter (EKF)-based solutions for computationally efficient joint estimation and tracking of the time of arrival (ToA) and direction of arrival (DoA) of the user nodes (UNs) using uplink reference signals. Then, a second EKF stage is proposed in order to fuse the individual DoA/ToA estimates from one or several ANs into a UN position estimate. Since all the processing takes place at the network side, the computing complexity and energy consumption at the UN side are kept to a minimum. The cascaded EKFs proposed in this article also take into account the unavoidable relative clock offsets between UNs and ANs, such that reliable clock synchronization of the access-link is obtained as a valuable by-product. The proposed cascaded EKF scheme is then revised and extended to more general and challenging scenarios where not only the UNs have clock offsets against the network time, but also the ANs themselves are not mutually synchronized in time. Finally, comprehensive performance evaluations of the proposed solutions on a realistic 5G network setup, building on the METIS project based outdoor Madrid map model together with complete ray tracing based propagation modeling, are provided. The obtained results clearly demonstrate that by using the developed methods, sub-meter scale positioning and tracking accuracy of moving devices is indeed technically feasible in future 5G radio access networks operating at sub-6GHz frequencies, despite the realistic assumptions related to clock offsets and potentially even under unsynchronized network elements.