This paper investigates semantic communications between a transmitter and a receiver, where original data, such as videos of interest to the receiver, is stored at the transmitter. Although significant process has been made in semantic communications, a fundamental design problem is that the semantic information is extracted based on certain criteria at the transmitter alone, without considering the receiver's specific information needs. As a result, critical information of primary concern to the receiver may be lost. In such cases, the semantic transmission becomes meaningless to the receiver, as all received information is irrelevant to its interests. To solve this problem, this paper presents a receiver-centric generative semantic communication system, where each transmission is initialized by the receiver. Specifically, the receiver first sends its request for the desired semantic information to the transmitter at the start of each transmission. Then, the transmitter extracts the required semantic information accordingly. A key challenge is how the transmitter understands the receiver's requests for semantic information and extracts the required semantic information in a reasonable and robust manner. We address this challenge by designing a well-structured framework and leveraging off-the-shelf generative AI products, such as GPT-4, along with several specialized tools for detection and estimation. Evaluation results demonstrate the feasibility and effectiveness of the proposed new semantic communication system.