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Generative Adversarial Networks (GANs) are Machine Learning (ML) algorithms that have the ability to address competitive resource allocation problems together with detection and mitigation of anomalous behavior. In this paper, we investigate their use in next-generation (NextG) communications within the context of cognitive networks to address i) spectrum sharing, ii) detecting anomalies, and iii) mitigating security attacks. GANs have the following advantages. First, they can learn and synthesize field data, which can be costly, time consuming, and nonrepeatable. Second, they enable pre-training classifiers by using semi-supervised data. Third, they facilitate increased resolution. Fourth, they enable the recovery of corrupted bits in the spectrum. The paper provides the basics of GANs, a comparative discussion on different kinds of GANs, performance measures for GANs in computer vision and image processing as well as wireless applications, a number of datasets for wireless applications, performance measures for general classifiers, a survey of the literature on GANs for i)-iii) above, and future research directions. As a use case of GAN for NextG communications, we show that a GAN can be effectively applied for anomaly detection in signal classification (e.g., user authentication) outperforming another state-of-the-art ML technique such as an autoencoder.