The objective of the dissertation is to bridge the gap between the theoretical study of covert capacity and the practical design of provably covert systems. Covert communications require the legitimate parties in a communication network to establish reliable transmission without being detected by malicious adversaries. Although the optimal number of covert information bits has been shown to grow in the square root of channel uses, such square-root law can be circumvented by exploring the information asymmetry between the legitimate parties and adversaries. This prompts the pursuit of improved covert throughput if one exploits resources or uncertainties of the communication channels and signals that are inaccessible to adversaries. Another subtlety is the constant behind the scaling of covert information bits, which plays the role of the covert capacity, exhibits a strong dependency on how the covertness is measured by mathematical metrics. In particular, we choose the variational distance as the metric throughout the dissertation, as it captures the operational meaning of adversaries' hypothesis tests. These two aspects constitute the motivating threads of the entire dissertation. In the first half of the dissertation, we investigate classical covert communications. In particular, we study the covert capacity in a multi-antenna system; we design a provably covert code via polar codes and invertible extractors; we explore the benefits of asynchronism in covert communication systems. In the second half of the dissertation, we extend our investigation to thermal-noise bosonic channels, which model quantum optical communications. We present a resource-efficient scheme to exhibit the benefits of quantum entanglement for covert communications and the trade-off between classical and quantum resources. Lastly, inspired by the recent development of joint communication and sensing systems, we characterize the information-theoretic limit of joint communication and sensing with a covertness constraint in terms of covert throughputs and sensing-error exponents.