Neural Networks have proven themselves very adept at solving a wide variety of problems, in particular they accel at image processing. However, it remains unknown how well they perform under memory errors. This thesis focuses on the robustness of neural networks under memory errors, specifically single event upset style errors where single bits flip in a network's trained parameters. The main goal of these experiments is to determine if different neural network architectures are more robust than others. Initial experiments show that MLPs are more robust than CNNs. Within MLPs, deeper MLPs are more robust and for CNNs larger kernels are more robust. Additionally, the CNNs displayed bimodal failure behavior, where memory errors would either not affect the performance of the network, or they would degrade its performance to be on par with random guessing. VGG16, ResNet50, and InceptionV3 were also tested for their robustness. ResNet50 and InceptionV3 were both more robust than VGG16. This could be due to their use of Batch Normalization or the fact that ResNet50 and InceptionV3 both use shortcut connections in their hidden layers. After determining which networks were most robust, some estimated error rates from neutrons were calculated for space environments to determine if these architectures were robust enough to survive. It was determined that large MLPs, ResNet50, and InceptionV3 could survive in Low Earth Orbit on commercial memory technology and only use software error correction.