This survey provides a comprehensive review of graph embedding, a technique that converts graph data into a low-dimensional space while preserving structural information. The survey begins by introducing the formal definition of graph embedding and related concepts. It then proposes two taxonomies of graph embedding based on problem settings and embedding techniques, highlighting the challenges and solutions in different scenarios. The survey also categorizes graph embedding applications into node-related, edge-related, and graph-related categories and suggests four promising future research directions in terms of computational efficiency, problem settings, solution techniques, and application scenarios. The contributions of the survey include a detailed analysis of graph embedding techniques, a systematic categorization of applications, and insights into the challenges and solutions in various problem settings. The survey covers a wide range of graph types, including homogeneous, heterogeneous, and graphs with auxiliary information, and discusses the challenges and techniques for each type. It also explores the use of deep learning and matrix factorization for graph embedding, providing a comprehensive overview of the field.This survey provides a comprehensive review of graph embedding, a technique that converts graph data into a low-dimensional space while preserving structural information. The survey begins by introducing the formal definition of graph embedding and related concepts. It then proposes two taxonomies of graph embedding based on problem settings and embedding techniques, highlighting the challenges and solutions in different scenarios. The survey also categorizes graph embedding applications into node-related, edge-related, and graph-related categories and suggests four promising future research directions in terms of computational efficiency, problem settings, solution techniques, and application scenarios. The contributions of the survey include a detailed analysis of graph embedding techniques, a systematic categorization of applications, and insights into the challenges and solutions in various problem settings. The survey covers a wide range of graph types, including homogeneous, heterogeneous, and graphs with auxiliary information, and discusses the challenges and techniques for each type. It also explores the use of deep learning and matrix factorization for graph embedding, providing a comprehensive overview of the field.