Machine learning models are increasingly used in many engineering fields thanks to the widespread digital data, growing computing power, and advanced algorithms. Artificial neural networks (ANN) is the most popular machine learning model in recent years. Although many ANN models have been used in the design and analysis of composite materials and structures, there are still some unsolved issues that hinder the acceptance of ANN models in the practical design and analysis of composite materials and structures. Moreover, the emerging machine learning techniques are posting new opportunities and challenges in the data-based design paradigm. This paper aims to give a state-of-the-art literature review of ANN models in the nonlinear constitutive modeling, multiscale surrogate modeling, and design optimization of composite materials and structures. This review has been designed to focus on the discussion of the general frameworks and benefits of ANN models to the above problems. Moreover, challenges and opportunities in each key problem are identified and discussed. This paper is expected to open the discussion of future research scope and new directions to enable efficient, robust, and accurate data-driven design and analysis of composite materials and structures.