The concept of physical layer security builds on the pivotal idea of turning the channel's imperfections, such as noise and fading, into a source of security. This is established through appropriately designed coding techniques and signal processing strategies. In this vein, it has been shown that fading channels can enhance the transmission of confidential information and that a secure communication can be achieved even when the channel to the eavesdropper is better than the main channel. However, to fully benefit from what fading has to offer, the knowledge of the channel state information at the transmitter (CSIT) is of primordial importance. In practical wireless communication systems, CSIT is usually obtained, prior to data transmission, through CSI feedback sent by the receivers. The channel links over which this feedback information is sent can be either noisy, rate-limited, or delayed, leading to CSIT uncertainty. In this paper, we present a comprehensive review of recent and ongoing research works on physical layer security with CSIT uncertainty. We focus on both information theoretic and signal processing approaches to the topic when the uncertainty concerns the channel to the wiretapper or the channel to the legitimate receiver. Moreover, we present a classification of the research works based on the considered channel uncertainty. Mainly, we distinguish between the cases when the uncertainty comes from an estimation error of the CSIT, from a CSI feedback link with limited capacity, or from an outdated CSI.