Brain-inspired computing, drawing inspiration from the fundamental structure and information-processing mechanisms of the human brain, has gained significant momentum in recent years. It has emerged as a research paradigm centered around brain-computer dual-driven and multi-network integration. One noteworthy instance of this paradigm is the Hybrid Neural Network (HNN), which integrates computer-science-oriented artificial neural networks with neuroscience-oriented spiking neural networks. HNNs exhibit distinct advantages in various intelligent tasks, including perception, cognition, and learning. This paper presents a comprehensive review of HNNs with an emphasis on their origin, concepts, biological perspective, construction framework, and supporting systems. Furthermore, insights and suggestions for potential research directions are provided aiming to propel the advancement of the HNN paradigm.