The Helmholtz-Hodge Decomposition (HHD) describes the decomposition of a flow field into its divergence-free and curl-free components. Various communities like weather modeling, seismology, oceanology, and computer graphics are interested in understanding the incompressibility and vorticity of flow. The HHD has proven to be an important tool in the analysis of fluids, making it one of the fundamental theorems in fluid dynamics. The recent advances in flow analysis have led to its application in a number of research communities such as visualization, topological analysis, imaging, and robotics. However, since the initial body of work, primarily in the physics communities, research on the topic has become fragmented with different communities working largely in isolation often repeating and sometimes contradicting each others results. Additionally, different nomenclature has evolved which further obscures the fundamental connections between fields making the transfer of knowledge difficult. This survey attempts to address these problems by collecting a comprehensive list of relevant references and examining them using a common terminology. A particular focus is the discussion of boundary conditions of HHD. The goal is to promote further research in the field by creating a common repository of techniques to compute the HHD and a large collection of example applications in a broad range of areas.