Transmission through reconfigurable intelligent surfaces (RISs), which control the reflection/scattering characteristics of incident waves in a deliberate manner to enhance the signal quality at the receiver, appears as a promising candidate for future wireless communication systems. In this paper, we bring the concept of RIS-assisted communications to the realm of index modulation (IM) by proposing RIS-space shift keying (RIS-SSK) and RIS-spatial modulation (RIS-SM) schemes. These two schemes are realized through not only intelligent reflection of the incoming signals to improve the reception but also utilization of the IM principle for the indices of multiple receive antennas in a clever way to improve the spectral efficiency. Maximum energy-based suboptimal (greedy) and exhaustive search-based optimal (maximum likelihood) detectors of the proposed RIS-SSK/SM schemes are formulated and a unified framework is presented for the derivation of their theoretical average bit error probability. Extensive computer simulation results are provided to assess the potential of RIS-assisted IM schemes as well as to verify our theoretical derivations. Our findings also reveal that RIS-based IM, which enables high data rates with remarkably low error rates, can become a potential candidate for future wireless communication systems in the context of beyond multiple-input multiple-output (MIMO) solutions.