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The demand on wireless communications to provide high data rates, high mobility, and high quality of service poses more challenges for designers. To contend with deleterious channel fading effects, both the transmitter and the receiver must be designed appropriately to exploit the diversity embedded in the channels. From the perspective of receiver design, the ultimate goal is to achieve both low complexity and high performance. In this article, we first summarize the complexity and performance of low-complexity receivers, including linear equalizers and decision feedback equalizers, and then we reveal the fundamental condition when LEs and DFEs collect the same diversity as the maximum-likelihood equalizer. Recently, lattice reduction techniques were introduced to enhance the performance of low-complexity equalizers without increasing the complexity significantly. Thus, we also provide a comprehensive review of LR-aided low-complexity equalizers and analyze their performance. Furthermore, we describe the architecture and initial results of a very-large-scale-integration implementation of an LR algorithm. [ABSTRACT FROM AUTHOR]