<p>Hybrid spectral-neural network (NN–SM) methods, which combine the high accuracy of spectral methods with the flexibility and learning capabilities of neural networks, are becoming powerful tools for solving differential equations. While numerous studies have demonstrated the potential of these methods, the rapidly evolving landscape of these hybrid techniques lacks a unified systematic overview. This article addresses this gap by presenting a systematic review of the progress, current applications, and challenges in the development and implementation of NN–SM. One significant contribution is the methodical classification of the various approaches that integrate spectral methods with neural network architectures. We analyze various neural network designs, spectral integration strategies, and training methodologies to identify the wide range of problems to which these hybrid methods have been effectively applied. Key developments highlighted include the use of physics-informed neural networks, accelerated training methods like extreme learning machines, and adaptive spectral techniques that enhance both training speed and accuracy. The review synthesizes findings from a comprehensive analysis of peer-reviewed literature to map the evolution of NN-SM, identifying trends in their application to areas such as fluid dynamics and heat transfer for both simple and complex differential equations. Furthermore, we identify persistent challenges, including the fair comparison of hybrid methods with traditional numerical methods and the difficulty of solving high-dimensional or multiscale problems. By structuring the current state of the art and pinpointing research gaps, this review provides a roadmap for future research to advance the accuracy, stability, and efficiency of NN–SM.</p>