  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.
