Generative AI is transforming enterprise application development by enabling machines to create content, code, and designs. These models, however, demand substantial computational power and data management. Cloud computing addresses these needs by offering infrastructure to train, deploy, and scale generative AI models. This review examines cloud services for generative AI, focusing on key providers like Amazon Web Services (AWS), Microsoft Azure, Google Cloud, IBM Cloud, Oracle Cloud, and Alibaba Cloud. It compares their strengths, weaknesses, and impact on enterprise growth. We explore the role of high-performance computing (HPC), serverless architectures, edge computing, and storage in supporting generative AI. We also highlight the significance of data management, networking, and AI-specific tools in building and deploying these models. Additionally, the review addresses security concerns, including data privacy, compliance, and AI model protection. It assesses the performance and cost efficiency of various cloud providers and presents case studies from healthcare, finance, and entertainment. We conclude by discussing challenges and future directions, such as technical hurdles, vendor lock-in, sustainability, and regulatory issues. Put together, this work can serve as a guide for practitioners and researchers looking to adopt cloud-based generative AI solutions, serving as a valuable guide to navigating the intricacies of this evolving field.