Graph processing has become an important part of multiple areas of computer science, such as machine learning, computational sciences, medical applications, social network analysis, and many others. Numerous graphs such as web or social networks may contain up to trillions of edges. Often, these graphs are also dynamic (their structure changes over time) and have domain-specific rich data associated with vertices and edges. Graph database systems such as Neo4j enable storing, processing, and analyzing such large, evolving, and rich datasets. Due to the sheer size of such datasets, combined with the irregular nature of graph processing, these systems face unique design challenges. To facilitate the understanding of this emerging domain, we present the first survey and taxonomy of graph database systems. We focus on identifying and analyzing fundamental categories of these systems (e.g., triple stores, tuple stores, native graph database systems, or object-oriented systems), the associated graph models (e.g., RDF or Labeled Property Graph), data organization techniques (e.g., storing graph data in indexing structures or dividing data into records), and different aspects of data distribution and query execution (e.g., support for sharding and ACID). 51 graph database systems are presented and compared, including Neo4j, OrientDB, or Virtuoso. We outline graph database queries and relationships with associated domains (NoSQL stores, graph streaming, and dynamic graph algorithms). Finally, we describe research and engineering challenges to outline the future of graph databases.