A multilayered graph is a dispensable data representation tool to comprehend and mine the richness and complexity of complex systems in real-world scenarios. Multifaceted interconnecting relationships cast in multiple layers of the graph construct a holistic, versatile, and powerful framework that enables researchers to effectively process information across various domains. Regardless of the complex nature of graph-structured data, graph embedding techniques can reduce them into lower-dimensional vectors preserving the most valuable information representing their topology and properties. The proper choice of embedding method depends on several factors including graph characteristics, task specificities, learning setting restrictions, and available computational resources. This survey provides a comprehensive overview of the multilayered graph embedding method in data mining and machine learning fields. We also propose a taxonomy to study the state-of-the-art embedding methods grouped into three categories: algorithmic methods, machine learning methods, and deep learning methods. In order to support readers in different applications, a comprehensive review is derived to summarize, analyze, and compare embedding methods within and between those categories to support possible applications. Finally, we propose multiple potential research directions in this burgeoning field.