Widely-deployed encryption-based security prevents unauthorized decoding, but does not ensure undetectability of communication. However, covert, or low probability of detection/intercept (LPD/LPI) communication is crucial in many scenarios ranging from covert military operations and the organization of social unrest, to privacy protection for users of wireless networks. In addition, encrypted data or even just the transmission of a signal can arouse suspicion, and even the most theoretically robust encryption can often be defeated by a determined adversary using non-computational methods such as side-channel analysis. Various covert communication techniques were developed to address these concerns, including steganography for finite-alphabet noiseless applications and spread-spectrum systems for wireless communications. After reviewing these covert communication systems, this article discusses new results on the fundamental limits of their capabilities, as well as provides a vision for the future of such systems.