This paper provides a comprehensive review of analog memristor emulators (Memulators), starting with the theoretical foundation of memristors, historical development, and their physical structures. It then investigates their design methodologies and operational principles in various circuit topologies, encompassing both MOS-based and Active Building Block (ABB)-based designs, which are critically examined, highlighting their unique characteristics, performance trade-offs (including power consumption, layout area, and maximum operating frequency), and inherent design considerations. Addressing the lack of standardisation in the field, a concrete, minimal benchmark suite is proposed, specifying test signals and robustness tests to ensure reproducible characterisation. A detailed comparative analysis of recent analog memulator circuits is presented, employing a defined Figure of Merit (FoM) and introducing a novel framework of qualitative comparison tables. Unlike traditional performance listings, this framework meticulously evaluates designs based on validation rigour (distinguishing simulation from experimental silicon), architectural versatility, and electronic tunability, providing a systematic metric for practical deployment readiness. Furthermore, a critical analysis of implementation challenges — including sneak paths, device variability, and forming-free operation — synthesises how different topologies mitigate these issues. Key case studies of fabricated memulator designs are also highlighted, illustrating practical realisation and system-level application. The paper further explores a wide range of applications, with particular emphasis on neuromorphic computing and chaotic oscillators for secure communications, demonstrating their practical significance in modern electronics. Finally, recent advancements and innovations from the literature are summarised, alongside a thorough discussion of prevailing research trends and promising future research directions aimed at overcoming existing limitations and unlocking new capabilities.
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