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In this paper, we propose a new paradigm of fault diagnosis in dynamic systems as an alternative to the well-established observer-based framework. The basic idea behind this work is to (i) formulate fault detection and isolation as projection of measurement signals onto (system) subspaces in Hilbert space, and (ii) solve the resulting problems by means of projection methods with orthogonal projection operators and gap metric as major tools. In the new framework, fault diagnosis issues are uniformly addressed both in the model-based and data-driven fashions. Moreover, the design and implementation of the projection-based fault diagnosis systems, from residual generation to threshold setting, can be unifiedly handled. Thanks to the well-defined distance metric for projections in Hilbert subspaces, the projection-based fault diagnosis systems deliver optimal fault detectability. In particular, a new type of residual-driven thresholds is proposed, which significantly increases the fault detectability. In this work, various design schemes are proposed, including a basic projection-based fault detection scheme, fault detection schemes for feedback control systems, fault classification as well as two modified fault detection schemes. As a part of our study, relations to the existing observer-based fault detection systems are investigated, which showcases that, with comparable online computations, the proposed projection-based detection methods offer improved detection performance.