The development of generative AI models has understandably caused considerable excitement among machine learning professionals. Few have paid attention to the potential copyright implications of using massive amounts of data publicly available on the Internet to train these models. Commercial developers in the U.S. have expressed confidence that the copyright doctrine of fair use would shield them from liability. In the EU, recently adopted text and data mining exceptions seemed to legalize generative AI training. Israel and Japan have similar rules. But with more than forty copyright-related lawsuits pending against the largest generative AI developers in the U.S. and now a few in Canada, and with the EU and UK aiming to require compliance with their laws, copyright is looming large in the future of generative AI developers. While it is seemingly impossible to create a global licensing regime that would cover all uses of all in-copyright works as training data, proposals to establish collective licensing regimes are under discussion in the EU, UK, and U.S. The machine learning community needs to understand enough about these copyright debates to participate meaningfully in shaping legal environments that will foster innovation in this field, support scientific research, create socially valuable tools, and treat works and their authors with respect.