Graphs are a tremendously suitable data representation that models the relationships of entities in many application domains, such as recommendation systems, machine learning, computational biology, social network analysis, and other application domains. Graphs with many vertices and edges have become quite prevalent in recent years. Therefore, graph computing systems with integrated various graph partitioning techniques have been envisioned as a promising paradigm to handle large-scale graph analytics in these application domains. However, scalable processing of large-scale graphs is challenging due to their high volume and inherent irregular structure of the real-world graphs. Hence, industry and academia have recently proposed graph partitioning and computing systems to efficiently process and analyze large-scale graphs. The graph partitioning and computing systems have been designed to improve scalability issues and reduce processing time complexity. This paper presents an overview, classification, and investigation of the most popular graph partitioning and computing systems. The various methods and approaches of graph partitioning and diverse categories of graph computing systems are presented. Finally, we discuss future challenges and research directions in graph partitioning and computing systems.