The rapid advancement of artificial intelligence technologies has given rise to diversified intelligent services, which place unprecedented demands on massive connectivity and gigantic data aggregation. However, the scarce radio resources and stringent latency requirement make it challenging to meet these demands. To tackle these challenges, over-the-air computation (AirComp) emerges as a potential technology. Specifically, AirComp seamlessly integrates the communication and computation procedures through the superposition property of multiple-access channels, which yields a revolutionary multiple-access paradigm shift from “compute-after-communicate” to “compute-when-communicate”. By this means, AirComp enables spectral-efficient and low-latency wireless data aggregation by allowing multiple devices to occupy the same channel for transmission. In this paper, we aim to present the recent advancement of AirComp in terms of foundations, technologies, and applications. The mathematical form and communication design are introduced as the foundations of AirComp, and the critical issues of AirComp over different network architectures are then discussed along with the review of existing literature. The technologies employed for the analysis and optimization on AirComp are reviewed from the information theory and signal processing perspectives. Moreover, we present the existing studies that tackle the practical implementation issues in AirComp systems, and elaborate the applications of AirComp in Internet of Things and edge intelligent networks. Finally, potential research directions are highlighted to motivate the future development of AirComp.