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Purpose: Medical images acquired using different scanners and protocols can differ substantially in their appearance. This phenomenon, scanner domain shift, can result in a drop in the performance of deep neural networks which are trained on data acquired by one scanner and tested on another. This significant practical issue is well-acknowledged, however, no systematic study of the issue is available across different modalities and diagnostic tasks. Materials and Methods: In this paper, we present a broad experimental study evaluating the impact of scanner domain shift on convolutional neural network performance for different automated diagnostic tasks. We evaluate this phenomenon in common radiological modalities, including X-ray, CT, and MRI. Results: We find that network performance on data from a different scanner is almost always worse than on same-scanner data, and we quantify the degree of performance drop across different datasets. Notably, we find that this drop is most severe for MRI, moderate for X-ray, and quite small for CT, on average, which we attribute to the standardized nature of CT acquisition systems which is not present in MRI or X-ray. We also study how injecting varying amounts of target domain data into the training set, as well as adding noise to the training data, helps with generalization. Conclusion: Our results provide extensive experimental evidence and quantification of the extent of performance drop caused by scanner domain shift in deep learning across different modalities, with the goal of guiding the future development of robust deep learning models for medical image analysis.