Information Retrieval (IR) systems are crucial tools for users to access information, which have long been dominated by traditional methods relying on similarity matching. With the advancement of pre-trained language models, generative information retrieval (GenIR) emerges as a novel paradigm, attracting increasing attention. Based on the form of information provided to users, current research in GenIR can be categorized into two aspects: \textbf{(1) Generative Document Retrieval} (GR) leverages the generative model's parameters for memorizing documents, enabling retrieval by directly generating relevant document identifiers without explicit indexing. \textbf{(2) Reliable Response Generation} employs language models to directly generate information users seek, breaking the limitations of traditional IR in terms of document granularity and relevance matching while offering flexibility, efficiency, and creativity to meet practical needs. This paper aims to systematically review the latest research progress in GenIR. We will summarize the advancements in GR regarding model training and structure, document identifier, incremental learning, etc., as well as progress in reliable response generation in aspects of internal knowledge memorization, external knowledge augmentation, etc. We also review the evaluation, challenges and future developments in GenIR systems. This review aims to offer a comprehensive reference for researchers, encouraging further development in the GenIR field. Github Repository: https://github.com/RUC-NLPIR/GenIR-Survey