The performance of perovskite solar cells has increased at an unprecedented rate, with efficiencies currently exceeding 20%. This technology is particularly promising, as it is compatible with cheap solution processing. For a thin-film solar product to be commercially viable, it must pass the IEC 61646 testing standards, regarding the environmental stability. Currently, the poor stability of perovskite solar cells is a barrier to commercialisation. The main issue causing this problem is the instability of the perovskite layer when in contact with moisture; however, it is important to explore stability problems with the other layers and interfaces within the device. The stability issues discussed in this review highlight the need to view the device as a whole system, due to the interdependent relationships between the layers, including: the perovskite absorber, electron transport layers, hole transport layers, other buffer layers and the electrodes. We also discuss other issues pertaining to device stability, such as measurement-induced hysteresis and the requirement for standard testing protocols. For perovskite solar cells to achieve the required stability, future research must focus on improving the intrinsic stability of the perovskite absorber layer, carefully designing the device geometry, and finding durable encapsulant materials, which seal the device from moisture.