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Emergent wireless communication standards, which are employed in different transmission environments, support various modulation schemes. High-order constellations are targeted to achieve high bandwidth efficiency. However, the complexity of the symbol-by-symbol Maximum A Posteriori (MAP) algorithm increases dramatically for these high-order modulation schemes. In order to reduce the hardware complexity, the suboptimal Max-Log-MAP, which is the direct transformation of the MAP algorithm into logarithmic domain, is alternatively implemented. In the literature, a great deal of research effort has been invested into Max-Log-MAP demapping. Several simplifications are presented to meet with specific constellations. In addition, the hardware implementations dedicated for Max-Log-MAP demapping vary greatly in terms of design choices, supported flexibility and performance criteria, making them a challenge to compare. This paper explores the published Max-Log-MAP algorithm simplifications and existing hardware demapper designs and presents an extensive review of the current literature. In-depth comparisons are drawn amongst the designs and different key performance characteristics are described, namely, achieved throughput, hardware resource requirements and flexibility. This survey should facilitate fair comparisons of future designs, as well as opportunities for improving the design of Max-Log-MAP demappers. [ABSTRACT FROM AUTHOR]