In contemporary statistical learning, leveraging information across related datasets has emerged as a powerful tool to enhance inference and prediction. This thesis explores the theoretical foundations and practical methodologies of transfer learning within a rigorous statistical framework. We investigate three fundamental problems—functional mean estimation, covariance matrix estimation, and functional linear regression—to elucidate how information from auxiliary domains can be systematically harnessed to improve performance in a primary target domain. Across these problems, the thesis identifies the regimes in which transfer learning outperforms conventional methods and develops adaptive algorithms that are both theoretically sound and practically implementable. By establishing a systematic foundation for transfer learning in statistics, this work advances both the theoretical understanding and the methodological toolkit for transcending traditional data boundaries.