Traditional approaches to wireless communication security (e.g., encryption) focus on maintaining the message integrity so that the contents are only accessible to the intended recipient. However, detection of the mere presence of a transmission can have a negative impact, violating the privacy of the communicating parties. In contrast, Covert Communications (also known as Low Probability of Detection Communications) hide the transmission of a message from a watchful adversary while ensuring a certain decoding performance at the receiver. In this thesis, we focus on exploiting any existing or induced uncertainties at the adversary, developing novel methods to achieve covertness in wireless scenarios. The insights gained from this thesis aim to help alleviate the ever increasing security and privacy concerns in future wireless networks. The first half of the thesis examines the use of artificial noise (AN) to cause sufficient confusion at the adversary such that message transmissions cannot be detected. We first consider a full-duplex information receiver, who generates AN of varying power causing uncertainty in the adversary's received signal statistics. Although the transmission of this AN causes self-interference, it provides the opportunity of achieving covertness under carefully managed transmit power levels. Here, we provide design guidelines for the choice of AN transmission power range. Furthermore, we demonstrate that if the transmission probability and AN power can be jointly optimized, the prior transmission probability of 0.5, which amounts to a random guess by the adversary, is not always the best choice for achieving maximum covertness. Rather, increasing the transmission probability beyond 0.5 allows an increase in the AN transmit power for satisfying a given covert rate requirement and can be the difference between achieving strong covertness and almost no covertness at all. Relying on the use of AN, we next consider achieving covertness in the domain of backscatter radio systems. We assume that the tag (containing the information) is passive and the reader (transceiver) controls the transmit power to keep the tag's response hidden. A non-conventional transmission scheme is proposed where the reader emits noise-like signal with transmit power varying across different communication slots. We analyse the conditions on the transmit power to achieve a target level of covertness, and illustrate the price a backscatter system has to pay, in terms of bit error rate, for achieving covert communication. In the second half of the thesis, we focus on scenarios where users suffer from uncertainty in their channel knowledge under quasi-static fading conditions. We first focus on the case where the adversary can make an infinite number of observations in a time slot, and a public action is used to provide cover for a secret action. It has been demonstrated that although channel uncertainty adversely effects the information at the intended receiver, it also provides the opportunity to hide any transmissions. Secondly, under a finite blocklength assumption, we investigate Willie's optimal detection performance in two extreme cases, i.e., the case of perfect channel state information (CSI) and the case of channel distribution information (CDI) only. It is shown that in the large detection error regime, Willie's detection performances in these two extreme cases are essentially indistinguishable, implying that the quality of CSI does not help Willie in improving his detection performance. We, thus, reveal fundamental differences in the design of covert transmissions for quasi-static fading channels in comparison to non-fading AWGN channels.