With wide deployment of machine learning (ML)-based systems for a variety of applications including medical, military, automotive, genomic, multimedia, and social networking, there is great potential for damage from adversarial learning (AL) attacks. In this article, we provide a contemporary survey of AL, focused particularly on defenses against attacks on deep neural network classifiers. After introducing relevant terminology and the goals and range of possible knowledge of both attackers and defenders, we survey recent work on test-time evasion (TTE), data poisoning (DP), backdoor DP, and reverse engineering (RE) attacks and particularly defenses against the same. In so doing, we distinguish robust classification from anomaly detection (AD), unsupervised from supervised, and statistical hypothesis-based defenses from ones that do not have an explicit null (no attack) hypothesis. We also consider several scenarios for detecting backdoors. We provide a technical assessment for reviewed works, including identifying any issues/limitations, required hyperparameters, needed computational complexity, as well as the performance measures evaluated and the obtained quality. We then delve deeper, providing novel insights that challenge conventional AL wisdom and that target unresolved issues, including: robust classification versus AD as a defense strategy; the belief that attack success increases with attack strength, which ignores susceptibility to AD; small perturbations for TTE attacks: a fallacy or a requirement; validity of the universal assumption that a TTE attacker knows the ground-truth class for the example to be attacked; black, gray, or white-box attacks as the standard for defense evaluation; and susceptibility of query-based RE to an AD defense. We also discuss attacks on the privacy of training data. We then present benchmark comparisons of several defenses against TTE, RE, and backdoor DP attacks on images. The article concludes with a discussion of continuing research directions, including the supreme challenge of detecting attacks whose goal is not to alter classification decisions, but rather simply to embed, without detection, "fake news" or other false content.