Model-based monitoring determines faults in a supervised system by comparing the available system's measurements with a priori information represented by the system's mathematical model. Especially in technical environments, a monitoring system must be able to reason with incomplete knowledge about the supervised system, to process noisy and erroneous observations and to react within a limited time. We present MOSES, a model-based monitoring system which is based on imprecise models where the structure is known and the parameters may be imprecisely specified by numerical intervals. As a consequence, only bounds on the trajectories can be derived with imprecise models. These bounds are computed using traditional numerical integration techniques starting from individual points on the external surface of the model's uncertainty space. When new measurements from the supervised system become available, MOSES checks the consistency of this new information with the model's prediction and refutes inconsistent parts from the uncertainty space of the model. A fault in the supervised system is detected when the complete model's uncertainty space has been refuted. MOSES bridges and extends methodologies from the FDI and DX communities by refining the model's uncertainty space conservatively through refutation, by applying standard numerical techniques for deriving the trajectories of imprecise models and by exploiting the measurements as soon as possible for online monitoring. The performance of MOSES is evaluated based on examples and by online monitoring a complex heating system.