This state of the art report covers reconstruction methods for transparent and specular objects or phenomena. While the 3D acquisition of opaque surfaces with lambertian reflectance is a well-studied problem, transparent, refractive, specular and potentially dynamic scenes pose challenging problems for acquisition systems. This report reviews and categorizes the literature in this field. Despite tremendous interest in object digitization, the acquisition of digital models of transparent or specular objects is far from being a solved problem. On the other hand, real-world data is in high demand for applications such as object modeling, preservation of historic artifacts and as input to data driven modeling techniques. With this report we aim at providing a reference for and an introduction to the field of transparent and specular object reconstruction. We describe acquisition approaches for different classes of objects. Transparent objects/phenomena that do not change the straight ray geometry can be found foremost in natural phenomena. Refraction effects are usually small and can be considered negligible for these objects. Phenomena as diverse as fire, smoke, and interstellar nebulae can be modeled using a straight ray model of image formation. Refractive and specular surfaces on the other hand change the straight rays into usually piecewise linear ray paths, adding additional complexity to the reconstruction problem. Translucent objects exhibit significant sub-surface scattering effects rendering traditional acquisition approaches unstable. Different classes of techniques have been developed to deal with these problems and good reconstruction results can be achieved with current state-of-the-art techniques. However, the approaches are still specialized and targeted at very specific object classes. We classify the existing literature and hope to provide an entry point to this exiting field.