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Rain removal aims to remove rain streaks from images/videos and reduce the disruptive effects caused by rain. It not only enhances image/video visibility but also allows many computer vision algorithms to function properly. This paper makes the first attempt to conduct a comprehensive study on the robustness of deep learning-based rain removal methods against adversarial attacks. Our study shows that, when the image/video is highly degraded, rain removal methods are more vulnerable to the adversarial attacks as small distortions/perturbations become less noticeable or detectable. In this paper, we first present a comprehensive empirical evaluation of various methods at different levels of attacks and with various losses/targets to generate the perturbations from the perspective of human perception and machine analysis tasks. A systematic evaluation of key modules in existing methods is performed in terms of their robustness against adversarial attacks. From the insights of our analysis, we construct a more robust deraining method by integrating these effective modules. Finally, we examine various types of adversarial attacks that are specific to deraining problems and their effects on both human and machine vision tasks, including 1) rain region attacks, adding perturbations only in the rain regions to make the perturbations in the attacked rain images less visible; 2) object-sensitive attacks, adding perturbations only in regions near the given objects. Code is available at https://github.com/yuyi-sd/Robust_Rain_Removal.