In the era of data-centric computing, the quantity of data is expected to increase exponentially. The physical separation of memory and processing units in traditional computers results in a considerable amount of unnecessary energy loss and time delay in the process of data calculation and storage. Devices based on ferroelectric materials possess the advantage of integrated data storage and computing. Nevertheless, research in the field of advanced computing has been constrained due to the incompatibility of traditional ferroelectrics (e.g., perovskites) with complementary metal oxide semiconductor (CMOS) technology and poor scalability. In recent years, research and innovation in hafnium (Hf)-based ferroelectrics have reignited interest in this field. The inherent CMOS compatibility, high coercive field (Ec), and high energy band gap of Hf-based ferroelectrics make their devices highly suitable for data storage. Moreover, the negative capacitance field-effect transistor (NCFET) based on Hf-based ferroelectrics can be utilized as a representative logic computing device. In addition, the multi-level weights of biological synapses can be accurately simulated by adjusting the controllable multi-domain polarization switching in Hf-based ferroelectric films, which indicates that Hf-based ferroelectrics will also have general advantages in the field of neuromorphic computing. However, the basic mechanisms and research progress of Hf-based ferroelectrics in these advanced computing fields have not been systematically summarized and sorted out. In this paper, we summarize the latest research results of Hf-based ferroelectrics in advanced computing. We review the history of ferroelectric materials and the numerous advantages of Hf-based ferroelectrics, focusing on the working principles, research progress, and circuit applications of Hf-based ferroelectric logic and memory devices. Additionally, we review the basic concepts of neuromorphic computing, especially discussing the research progress of Hf-based ferroelectric neuromorphic devices and the circuit applications of hardware neural networks. Finally, we made a positive outlook on this field.