The construction industry is a vital sector of the global economy, but it faces many productivity challenges in various processes, such as design, planning, procurement, inspection, and maintenance. Generative artificial intelligence (AI), which can create novel and realistic data or content, such as text, image, video, or code, based on some input or prior knowledge, offers innovative and disruptive solutions to address these challenges. However, there is a gap in the literature on the current state, opportunities, and challenges of generative AI in the construction industry. This study aims to fill this gap by providing a state-of-the-art analysis of generative AI in construction, with three objectives: (1) to review and categorize the existing and emerging generative AI opportunities and challenges in the construction industry; (2) to propose a framework for construction firms to build customized generative AI solutions using their own data, comprising steps such as data collection, dataset curation, training custom large language model (LLM), model evaluation, and deployment; and (3) to demonstrate the framework via a case study of developing a generative model for querying contract documents. The results show that retrieval augmented generation (RAG) improves the baseline LLM by 5.2, 9.4, and 4.8% in terms of quality, relevance, and reproducibility. This study provides academics and construction professionals with a comprehensive analysis and practical framework to guide the adoption of generative AI techniques to enhance productivity, quality, safety, and sustainability across the construction industry.