The low-dimensional-model-based electromagnetic imaging is an emerging member of the big family of computational imaging, by which the low-dimensional models of underlying signals are incorporated into both data acquisition systems and reconstruction algorithms for electromagnetic imaging, in order to improve the imaging performance and break the bottleneck of existing electromagnetic imaging methodologies. Over the past decade, we have witnessed profound impacts of the low-dimensional models on electromagnetic imaging. However, the low-dimensional-model-based electromagnetic imaging remains at its early stage, and many important issues relevant to practical applications need to be carefully investigated. Especially, we are in the big-data era of booming electromagnetic sensing, by which massive data are being collected for retrieving very detailed information of probed objects. This survey gives a comprehensive overview on the low-dimensional models of structure signals, along with its relevant theories and low-complexity algorithms of signal recovery. Afterwards, we review the recent advancements of low-dimensional-model-based electromagnetic imaging in various applied areas. We hope this survey could bridge the gap between the model-based signal processing and the electromagnetic imaging, advance the development of low-dimensional-model-based electromagnetic imaging, and serve as a basic reference in the future research of the electromagnetic imaging across various frequency ranges.