Emerging Distributed AI systems are revolutionizing big data computing and data processing capabilities with growing economic and societal impact. However, recent studies have identified new attack surfaces and risks caused by security, privacy, and fairness issues in AI systems. In this paper, we review representative techniques, algorithms, and theoretical foundations for trustworthy distributed AI through robustness guarantee, privacy protection, and fairness awareness in distributed learning. We first provide a brief overview of alternative architectures for distributed learning, discuss inherent vulnerabilities for security, privacy, and fairness of AI algorithms in distributed learning, and analyze why these problems are present in distributed learning regardless of specific architectures. Then we provide a unique taxonomy of countermeasures for trustworthy distributed AI, covering (1) robustness to evasion attacks and irregular queries at inference, and robustness to poisoning attacks, Byzantine attacks, and irregular data distribution during training; (2) privacy protection during distributed learning and model inference at deployment; and (3) AI fairness and governance with respect to both data and models. We conclude with a discussion on open challenges and future research directions toward trustworthy distributed AI, such as the need for trustworthy AI policy guidelines, the AI responsibility-utility co-design, and incentives and compliance.