<p>The rapid advancement of generative AI (GenAI) has introduced transformative methodologies to the financial sector, enabling the creation of novel data and solutions to longstanding challenges such as data scarcity, privacy, and domain adaptation. Despite the proliferation of GenAI research in finance, there is a lack of comprehensive surveys that systematically review the core generative techniques and the unique research problems posed by different financial data modalities. This paper addresses this gap by providing an in-depth overview of foundational GenAI methodologies, including Variational Autoencoders (VAEs), Generative Adversarial Networks (GANs), Normalizing Flow, Diffusion Models, and Large Language Models (LLMs), as well as their adaptations for financial tasks. We categorize and analyze research challenges according to data modalities: textual, time series, tabular, and graph data. For each modality, we introduce representative tasks such as classification, forecasting, question answering, and synthetic data generation, and further discuss current limitations and future research directions. This survey aims to serve as a technical reference for researchers and practitioners seeking to understand and advance GenAI techniques in the financial domain.</p>