Title: Advancing Nearest Neighbor Explanation-by-Example with Critical Classification Regions. Abstract: There is an increasing body of evidence suggesting that post-hoc explanation-by- example with nearest neighbors is a promising solution for the eXplainable Artificial Intelligence (XAI) problem. However, despite being thoroughly researched for decades, such post-hoc methods have never seriously explored how to enhance these explanations by highlighting specific important "parts" in a classification. Here, we propose the notion of Critical Classification Regions (CCRs) to do this, and several possible methods are experimentally compared to determine the best approach for this explanation strategy. CCRs supplement nearest neighbor examples by highlighting similar important "parts" in the image explanation. Experiments across multiple domains show that CCRs represent key features used by the CNN in both the testing and training data. Finally, a suitably-controlled user study (N=163) on ImageNet, shows CCRs improve people’s assessments of the correctness of a CNN’s predictions for difficult classifications due to ambiguity.