As deep learning continues to dominate all state-of-the-art computer vision tasks, it is increasingly becoming an essential building block for robotic perception. This raises important questions concerning the safety and reliability of learning-based perception systems. There is an established field that studies safety certification and convergence guarantees of complex software systems at design-time. However, the unknown future deployment environments of an autonomous system and the complexity of learning-based perception make the generalization of design-time verification to run-time problematic. In the face of this challenge, more attention is starting to focus on run-time monitoring of performance and reliability of perception systems with several trends emerging in the literature. This paper attempts to identify these trends and summarise the various approaches to the topic.