The development of novel hardware computing systems and methods has been a topic of increased interest for researchers worldwide. New materials, devices, and architectures are being explored as a means to deliver more efficient solutions to contemporary issues. Along with the advancement of technology, there is a continuous increase in methods available to address significant challenges. However, the increased needs to be fulfilled have also led to problems of increasing complexity that require better and faster computing and processing capabilities. Moreover, there is a wide range of problems in several applications that cannot be addressed using the currently available methods and tools. As a consequence, the need for emerging and more efficient computing methods is of utmost importance and constitutes a topic of active research. Among several proposed solutions, we distinguish the development of a novel nanoelectronic device, called a "memristor", that can be utilized both for storing and processing, and thus it has emerged as a promising circuit element for the design of compact and energy-efficient circuits and systems. The memristor has been proposed for a wide range of applications. However, in this work, we focus on its use in computing architectures based on the concept of Cellular Automata. The combination of the memristor's performance characteristics with Cellular Automata has boosted further the concept of processing and storing information on the same physical units of a system, which has been extensively studied in the literature as it provides a very good candidate for the implementation of Cellular Automata computing with increased potential and improved characteristics, compared to traditional hardware implementations. In this context, this paper reviews the most recent advancements toward the development of Cellular-Automata-based computing coupled with memristor devices. Several approaches for the design of such novel architectures, called "Memristive Cellular Automata", exist in the literature. This extensive review provides a thorough insight into the most important developments so far, helping the reader to grasp all the necessary information, which is here presented in an organized and structured manner. Thus, this article aims to pave the way for further development in the field and to bring attention to technological aspects that require further investigation.