Optical neural networks leverage the unique properties of light propagation, interference, and diffraction to implement neural network functionalities, offering distinctive advantages including large bandwidth, low power consumption, parallel processing capability, and compact integration. These merits have led to their widespread applications in high-performance computing and artificial intelligence. Among various implementations, integrated optical neural networks (IONN) have emerged as a prominent research focus due to the superior integration density, cost-effectiveness, strong anti-noise capability, and excellent scalability offered by integrated photonic platforms. Here, we provide a comprehensive review of IONN development, fundamental computing principles, recent advances in two representative integrated platforms, and performance evaluation metrics across different architectures. Furthermore, we critically discuss future research directions and key challenges in this rapidly evolving field.