In the k-medoid problem, given a dataset P, we are asked to choose kpoints in P as the medoids. The optimal medoid set minimizes the average Euclidean distance between the points in P and their closest medoid. Finding the optimal k medoids is NP hard, and existing algorithms aim at approximate answers, i.e., they compute medoids that achieve a small, yet not minimal, average distance. Similarly in this paper, we also aim at approximate solutions. We consider, however, the continuous version of the problem, where the points in P move and our task is to maintain the medoid set on-the-fly (trying to keep the average distance small). To the best of our knowledge, this work constitutes the first attempt on continuous medoid queries. First, we consider centralized monitoring, where the points issue location updates whenever they move. A server processes the stream of generated updates and constantly reports the current medoid set. Next, we address distributed monitoring, where we assume that the data points have some computational capabilities, and they take over part of the monitoring task. In particular, the server installs adaptive filters (i.e., permissible spatial ranges, called safe regions) to the points, which report their location only when they move outside their filters. The distributed techniques reduce the frequency of location updates (and, thus, the network overhead and the server load), at the cost of a slightly higher average distance, compared to the centralized methods. Both our centralized and distributed methods do not make any assumption about the data moving patterns (e.g., velocity vectors, trajectories, etc) and can be applied to an arbitrary number of medoids k. We demonstrate the efficiency and efficacy of our techniques through extensive experiments.