Prognostic Health Management (PHM) is a predictive maintenance strategy, which is based on Condition Monitoring (CM) data and aims to predict the future states of machinery. The existing literature reports the PHM at two levels: methodological and applicative. From the methodological point of view, there are many publications and standards of a PHM system design. From the applicative point of view, many papers address the improvement of techniques adopted for realizing PHM tasks without covering the whole process. In these cases, most applications rely on a large amount of historical data to train models for diagnostic and prognostic purposes. Industries, very often, are not able to obtain these data. Thus, the most adopted approaches, based on batch and off-line analysis, cannot be adopted. In this paper, we present a novel framework and architecture that support the initial application of PHM from the machinery producers’ perspective. The proposed framework is based on an edge-cloud infrastructure that allows performing streaming analysis at the edge to reduce the quantity of the data to store in permanent memory, to know the health status of the machinery at any point in time, and to discover novel and anomalous behaviors. The collection of the data from multiple machines into a cloud server allows training more accurate diagnostic and prognostic models using a higher amount of data, whose results will serve to predict the health status in real-time at the edge. The so-built PHM system would allow industries to monitor and supervise a machinery network placed in different locations and can thus bring several benefits to both machinery producers and users. After a brief literature review of signal processing, feature extraction, diagnostics, and prognostics, including incremental and semi-supervised approaches for anomaly and novelty detection applied to data streams, a case study is presented. It was conducted on data collected from a test rig and shows the potential of the proposed framework in terms of the ability to detect changes in the operating conditions and abrupt faults and storage memory saving. The outcomes of our work, as well as its major novel aspect, is the design of a framework for a PHM system based on specific requirements that directly originate from the industrial field, together with indications on which techniques can be adopted to achieve such goals.