In location-based, mobile continual query (CQ) systems, two key measures of quality-of-service (QoS) are: freshness and accuracy. To achieve freshness, the CQ server must perform frequent query reevaluations. To attain accuracy, the CQ server must receive and process frequent position updates from the mobile nodes. However, it is often difficult to obtain fresh and accurate CQ results simultaneously, due to 1) limited resources in computing and communication and 2) fast-changing load conditions caused by continuous mobile node movement. Hence, a key challenge for a mobile CQ system is: How do we achieve the highest possible quality of the CQ results, in both freshness and accuracy, with currently available resources? In this paper, we formulate this problem as a load shedding one, and develop MobiQual-a QoS-aware approach to performing both update load shedding and query load shedding. The design of MobiQual highlights three important features. 1) Differentiated load shedding: We apply different amounts of query load shedding and update load shedding to different groups of queries and mobile nodes, respectively. 2) Per-query QoS specification: Individualized QoS specifications are used to maximize the overall freshness and accuracy of the query results. 3) Low-cost adaptation: MobiQual dynamically adapts, with a minimal overhead, to changing load conditions and available resources. We conduct a set of comprehensive experiments to evaluate the effectiveness of MobiQual. The results show that, through a careful combination of update and query load shedding, the MobiQual approach leads to much higher freshness and accuracy in the query results in all cases, compared to existing approaches that lack the QoS-awareness properties of MobiQual, as well as the solutions that perform query-only or update-only load shedding.