Backscatter Communication (BackCom), which is based on passive reflection and modulation of an incident radio-frequency (RF) wave, has emerged as a cutting-edge technological paradigm for self-sustainable Internet-of-things (IoT). Nevertheless, the contemporary BackCom systems are limited to short-range and low data rate applications only, thus rendering them insufficient on their own to support pervasive connectivity among the massive number of IoT devices. Meanwhile, wireless networks are rapidly evolving towards the smart radio paradigm. In this regard, reconfigurable intelligent surfaces (RISs) have come to the forefront to transform the wireless propagation environment into a fully controllable and customizable space in a cost-effective and energy-efficient manner. Targeting the sixth-generation (6G) horizon, we anticipate the integration of RISs into BackCom systems as a new frontier for enabling 6G IoT networks. In this article, for the first time in the open literature, we provide a tutorial overview of RIS-assisted BackCom (RIS-BackCom) systems. Specifically, we introduce the four different variants of RIS-BackCom and identify the potential improvements that can be achieved by incorporating RISs into BackCom systems. In addition, owing to the unrivaled effectiveness of non-orthogonal multiple access (NOMA), we present a case study on an RIS-assisted NOMA-enhanced BackCom system. Finally, we outline the way forward for translating this disruptive concept into real-world applications.