Prognostics and health management is an emerging engineering technology that has been applied to a large variety of engineering systems to improve system's reliability. However, existing prognostics approaches have been developed largely based upon specific applications and system models, thus possess limited general applicability. This paper presents a generic data-driven prognostics method, namely an adaptive self-cognizant dynamic system (ASDS) approach, that integrates adaptive system recognition with a general state space-based dynamic system model for remaining useful life (RUL) prediction. The developed approach formulates a statistical learning framework with three core attributes: 1) a state-space-based dynamic system approach for the system performance modeling in general, 2) a data-driven method to learn time-series degradation performance of an engineering system, and 3) a Bayesian technique for self-updating of data-driven models to adapt to the operational or environmental changes. With the developed ASDS approach, the prognostics technique can eliminate the dependence on system specific models and be adaptive to system performance changes due to degradation or variation of system operational conditions, thereby yielding accurate RUL predictions. The developed methodology is applied to two engineering case studies to demonstrate its effectiveness.