Management of graph structured data has important applications in several areas. Queries on such data sets are based on structural properties of the graphs, in addition to values of attributes. Answering such queries pose significant challenges, as reasoning about structural properties across graphs are typically intractable problems. This chapter provides an overview of the challenges in designing databases over graph datasets. Different application areas that use graph databases, pose their own unique set of challenges, making the task of designing a generic graph-oriented DBMS still an elusive goal. The purpose of this chapter is to provide a tutorial introduction to some of the major challenges of graph data management, survey some of the piecemeal solutions that have been proposed, and suggest an overall structure in which these different solutions can be meaningfully placed.