Recently, the vulnerability of deep image classification models to adversarial attacks has been investigated. However, such an issue has not been thoroughly studied for image-to-image tasks that take an input image and generate an output image (e.g., colorization, denoising, deblurring, etc.) This paper presents comprehensive investigations into the vulnerability of deep image-to-image models to adversarial attacks. For five popular image-to-image tasks, 16 deep models are analyzed from various standpoints such as output quality degradation due to attacks, transferability of adversarial examples across different tasks, and characteristics of perturbations. We show that unlike image classification tasks, the performance degradation on image-to-image tasks largely differs depending on various factors, e.g., attack methods and task objectives. In addition, we analyze the effectiveness of conventional defense methods used for classification models in improving the robustness of the image-to-image models.