As one of the key technologies to maintain the safety and reliability of stochastic degrading systems, remaining useful life (RUL) prediction, also known as prognostics, has been attached great importance in recent years. Particularly, with the rapid development of industrial 4.0 and internet-of-things (IoT), prognostics for stochastic degrading systems under big data have been paid much attention in recent years and various prognosis methods have been reported. However, there has not been a critical review particularly focused on the strengths and weaknesses of these methods to provoke the new ideas for the prognostics research. To fill this gap, facing the realistic demand of prognostics of stochastic degrading systems under the background of big data, this paper profoundly analyzes the basic research ideas, development trends, and common problems of various data-driven prognostics methods, mainly including statistical data-driven methods, machine learning (ML) based methods, hybrid prognostics of statistical data-driven methods and ML based methods. Particularly, this paper discusses the emerging topic of prognosis under incomplete big data and the possible opportunities in the future are highlighted. Through discussing the pros and cons of existing methods, we provide discussions on challenges and possible opportunities to steer the future development of prognostics for stochastic degrading systems under big data. While an exhaustive review on prognostics methods remains elusive, we hope that the perspectives and discussions in this paper can serve as a stimulus for new prognostics research in the era of big data.