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This book summarizes new directions in mathematics education research on proving at the university level, thereby providing contemporary extensions of the sub-fields of proof that Annie and John Selden introduced to the field. The chapters each describe an emerging new area of proof research, review the relevant findings in this area, present open research questions and the tools to address them. The book also discusses proof as a literary genre, and how students' feelings during the proof writing process can influence their behavior. The concluding chapter of the book reflects on new directions for research on proving. As such, this book provides mathematics educators, who have extensive experience researching proof, with an up-to-date review of the new methodologies and research questions with regard to proof, and young scholars, interested in proof, can use these chapters as primers on which they can build a research program.