<p>The p-median problem has a central importance in the context of location planning problems. An extended version of this problem is the capacitated p-median problem (CPMP) which is used for diverse applications in urban planning and medical care units location. Given its relevance and its practicality, in this paper we present a large neighborhood search (LNS) and a study of hyper-heuristics for the CPMP. We propose and analyze various destruction operators within the framework of LNS to efficiently explore diverse neighborhoods. An exact solver is used in the repair phase. Additionally, these operators are also used for the hyper-heuristics, which are high-level problem-independent solution approaches, to propose new low-level heuristics. We provide a comparison of the LNS and of the best performing hyper-heuristics with state-of-art approaches for this problem. The proposed solution methods provide a lower average GAP value compared to the state of the art and find better solutions for several instances. We also give a detailed analysis of the performance of the low-level heuristics and their impact on the instances with different structure. These results confirm the robust performance of hyper-heuristics for the CPMP, which have the advantage to allow the employment on similar scenarios with minor adjustments.</p>