This tutorial-style overview article examines the fundamental principles and methods of robustness, using wireless sensing and communication (WSC) as the narrative and exemplifying framework. First, we formalize the conceptual and mathematical foundations of robustness, highlighting the interpretations and relations across robust statistics, optimization, and machine learning. Key techniques, such as robust estimation and testing, distributionally robust optimization, and regularized and adversary training, are investigated. Together, the costs of robustness in system design, for example, the compromised nominal performances and the extra computational burdens, are discussed. Second, we review recent robust signal processing solutions for WSC that address model mismatch, data scarcity, adversarial perturbation, and distributional shift. Specific applications include robust ranging-based localization, modality sensing, channel estimation, receive combining, waveform design, and federated learning. Through this effort, we aim to introduce the classical developments and recent advances in robustness theory to the general signal processing community, exemplifying how robust statistical, optimization, and machine learning approaches can address the uncertainties inherent in WSC systems.