Heterogeneous tabular data are the most commonly used form of data and are essential for numerous critical and computationally demanding applications. On homogeneous datasets, deep neural networks have repeatedly shown excellent performance and have therefore been widely adopted. However, their adaptation to tabular data for inference or data generation tasks remains highly challenging. To facilitate further progress in the field, this work provides an overview of state-of-the-art deep learning methods for tabular data. We categorize these methods into three groups: data transformations, specialized architectures, and regularization models. For each of these groups, our work offers a comprehensive overview of the main approaches. Moreover, we discuss deep learning approaches for generating tabular data and also provide an overview over strategies for explaining deep models on tabular data. Thus, our first contribution is to address the main research streams and existing methodologies in the mentioned areas while highlighting relevant challenges and open research questions. Our second contribution is to provide an empirical comparison of traditional machine learning methods with 11 deep learning approaches across five popular real-world tabular datasets of different sizes and with different learning objectives. Our results, which we have made publicly available as competitive benchmarks, indicate that algorithms based on gradient-boosted tree ensembles still mostly outperform deep learning models on supervised learning tasks, suggesting that the research progress on competitive deep learning models for tabular data is stagnating. To the best of our knowledge, this is the first in-depth overview of deep learning approaches for tabular data; as such, this work can serve as a valuable starting point to guide researchers and practitioners interested in deep learning with tabular data.