The introduction of Quantum Theory (QT) provides a unified mathematical framework for Information Retrieval (IR). Compared with the classical IR framework, the quantum-inspired IR framework is based on user-centered modeling methods to model non-classical cognitive phenomena in human relevance judgment in the IR process. With the increase of data and computing resources, neural IR methods have been applied to the text matching and understanding task of IR. Neural networks have a strong learning ability of effective representation and generalization of matching patterns from raw data. This monograph provides a systematic introduction to quantum-inspired neural IR, including quantum-inspired neural language representation, matching and understanding. The cross-field research on QT, neural network and IR is not only helpful to non-classical phenomena modeling in IR but also to break the theoretical bottleneck of neural networks and design more transparent neural IR models. The authors first introduce the language representation method based on QT. Secondly, they introduce the quantum-inspired text matching and decision making model under neural network that shows its theoretical advantages in document ranking, relevance matching, multimodal IR, and can be integrated with neural network to jointly promote the development of IR. Finally, the latest progress of quantum language understanding is introduced and further topics on QT and language modeling provide readers with more materials for thinking.