Abstract: The continuum topology optimization method can extensively improve the structural performance from the mechanical essence, which can provide designers with a variety of innovative design candidates. Due to these advantages and significant help to engineering, the continuum topology optimization method has been rapidly developed in recent years. This field has been developed relatively mature in dealing with linear topology optimization design problems. And it has been successfully applied to the high-performance design of all sorts of engineering structures. However, a large number of nonlinear issues are inherently involved in practical engineering. If they are assumed as linear problems, significant errors will often generate, and even wrong results may be obtained. This may ultimately lead to substantial engineering safety accidents. Under the demand-driven background of important engineering fields such as aerospace, mechanical engineering, marine engineering, high-speed trains, and architectural engineering, nonlinear continuum topology optimization methods have made remarkable progress in recent years. This paper aims to systematically review three types of nonlinear continuum topology optimization methods involving material nonlinearity, geometric nonlinearity, and boundary nonlinearity, with the typical methods comprehensively discussed and reviewed. Finally, the current difficulties (e.g., the poor numerical analysis accuracy, the low computational efficiency, limited to the field of statics, etc.) and future development directions (e.g., the large deformation and large strain problems, the nonlinear dynamic problems, the large-scale topology optimization design problems, etc.) of nonlinear continuum topology optimization methods are highlighted. This research review can provide a comprehensive knowledge sorting for beginners in the field of nonlinear continuum topology optimization. Moreover, it can also provide due help for scholars engaged in nonlinear continuum topology optimization methods.