Relaying is an effective technique to achieve reliable wireless connectivity in harsh communication environment. However, most of the existing relaying schemes are based on relays with fixed locations, or \emph{static relaying}. In this paper, we consider a novel \emph{mobile relaying} technique, where the relay nodes are assumed to be capable of moving at high speed. Compared to static relaying, mobile relaying offers a new degree of freedom for performance enhancement via careful relay trajectory design. We study the throughput maximization problem in mobile relaying systems by optimizing the source/relay transmit power along with the relay trajectory, subject to practical mobility constraints (on the relay speed and initial/final relay locations), as well as the \emph{information-causality constraint} at the relay owing to its decode-store-and-forward (DSF) strategy. It is shown that for fixed relay trajectory, the throughput-optimal source/relay power allocations over time follow a "staircase" water filling (WF) structure, with \emph{non-increasing} and \emph{non-decreasing} water levels at the source and relay, respectively. On the other hand, with given power allocations, the throughput can be further improved by optimizing the relay trajectory via successive convex optimization. An iterative algorithm is thus proposed to optimize the power allocations and relay trajectory alternately. Furthermore, for the special case with free initial and final relay locations, the jointly optimal power allocation and relay trajectory are derived. Numerical results show that by optimizing the trajectory of the relay and power allocations adaptive to its induced channel variation, mobile relaying is able to achieve significant throughput gains over the conventional static relaying.