Reconfigurable Intelligent Surfaces (RISs) are expected to play a pivotal role in future indoor ultra high data rate wireless communications as well as highly accurate three-dimensional localization and sensing, mainly due to their capability to provide flexible, cost- and power-efficient coverage extension, even under blockage conditions. However, when considering beyond millimeter wave frequencies where there exists GHz-level available bandwidth, realistic models of indoor RIS-parameterized channels verified by field-trial measurements are unavailable. In this article, we first present and characterize three RIS prototypes with unit cells of half-wavelength intercell spacing, which were optimized to offer a specific nonspecular reflection with 1-, 2-, and 3-bit phase quantization at 304 GHz. The designed static RISs were considered in an indoor channel measurement campaign carried out with a 304 GHz channel sounder. Channel measurements for two setups, one focusing on the transmitter-RIS-receiver path gain and the other on the angular spread of multipath components, are presented and compared with both state-of-the-art theoretical models as well as full-wave simulation results. The article is concluded with a list of challenges and research directions for RIS design and modeling of RIS-parameterized channels at THz frequencies.