A central topic in scientific visualization is the transfer function (TF) for volume rendering. The TF serves a fundamental role in translating scalar and multivariate data into color and opacity to express and reveal the relevant features present in the data studied. Beyond this core functionality, TFs also serve as a tool for encoding and utilizing domain knowledge and as an expression for visual design of material appearances. TFs also enable interactive volumetric exploration of complex data. The purpose of this state-of-the-art report (STAR) is to provide an overview of research into the various aspects of TFs, which lead to interpretation of the underlying data through the use of meaningful visual representations. The STAR classifies TF research into the following aspects: dimensionality, derived attributes, aggregated attributes, rendering aspects, automation, and user interfaces. The STAR concludes with some interesting research challenges that form the basis of an agenda for the development of next generation TF tools and methodologies.