Hybrid inorganic-organic perovskites have quickly evolved as a promising group of materials for solar cells and optoelectronic applications mainly owing to the inexpensive materials, relatively simple and versatile fabrication and high power conversion efficiency (PCE). The certified energy conversion efficiency for perovskite solar cell (PSC) has reached above 20%, which is compatible to the current best for commercial applications. However, long-term stabilities of the materials and devices remain to be the biggest challenging issue for realistic implementation of the PSCs. This article discusses the key issues related to the stability of perovskite absorbing layer including crystal structural stability, chemical stability under moisture, oxygen, illumination and interface reaction, effects of electron-transporting materials (ETM), hole-transporting materials (HTM), contact electrodes and preparation conditions. Towards the end, prospective strategies for improving the stability of PSCs are also briefly discussed and summarized. We focus on recent understanding of the stability of materials and devices and our perspectives about the strategies for the stability improvement.