Relay deployment in future mobile networks is a vital measure to enhance the coverage region of regular base stations, to overcome shadowing dips, and to bring improvements in the cell-edge performance. In this regard, resource allocation in a relay-enhanced scenario is a key design task, and has become a very interesting research topic over the past few years. In this article, we study two main resource allocation aspects of a relay-enhanced scenario. First, we concentrate on the problem of multiplexing the relay backhaul link and the direct link at the base station scheduler for in-band as well as out-band relay operations. We propose three distinct resource partitioning strategies, and evaluate their performance via long-term evolution (LTE) system level simulations in the downlink direction. We observe that even the low implementation effort algorithms bring appreciable improvement for the cell-edge users with or without small loss in performance for the other users, thereby enhancing system fairness. Second, we visit the problem of supporting heterogeneous quality of service (QoS) requirements in a multi-user relay-enhanced network. To this end, we introduce a QoS-aware scheduler which uses packet latency and rate requirements to prioritise the scheduling decisions. Moreover, we also propose a mechanism to support QoS-constrained services to the relayed users, served over two (or more) hops. Employing an LTE system level simulator, for relay-enhanced scenario with a traffic mix having distinct QoS requirements, we demonstrate that the proposed QoS-aware resource allocation strategy significantly increases the fraction of QoS-satisfied users.