The application of transfer learning, leveraging knowledge from source domains to enhance model performance in a target domain, has significantly grown, supporting diverse real-world applications. Its success often relies on shared knowledge between domains, typically required in these methodologies. Commonly, methods assume identical feature and label spaces in both domains, known as homogeneous transfer learning. However, this is often impractical as source and target domains usually differ in these spaces, making precise data matching challenging and costly. Consequently, heterogeneous transfer learning (HTL), which addresses these disparities, has become a vital strategy in various tasks. In this paper, we offer an extensive review of over 60 HTL methods, covering both data-based and model-based approaches. We describe the key assumptions and algorithms of these methods and systematically categorize them into instance-based, feature representation-based, parameter regularization, and parameter tuning techniques. Additionally, we explore applications in natural language processing, computer vision, multimodal learning, and biomedicine, aiming to deepen understanding and stimulate further research in these areas. Our paper includes recent advancements in HTL, such as the introduction of transformer-based models and multimodal learning techniques, ensuring the review captures the latest developments in the field. We identify key limitations in current HTL studies and offer systematic guidance for future research, highlighting areas needing further exploration and suggesting potential directions for advancing the field.