The influx of deep learning (DL) techniques into the field of survival analysis in recent years has led to substantial methodological progress; for instance, learning from unstructured or high-dimensional data such as images, text or omics data. In this work, we conduct a comprehensive systematic review of DL-based methods for time-to-event analysis, characterizing them according to both survival- and DL-related attributes. In summary, the reviewed methods often address only a small subset of tasks relevant to time-to-event data - e.g., single-risk right-censored data - and neglect to incorporate more complex settings. Our findings are summarized in an editable, open-source, interactive table: https://survival-org.github.io/DL4Survival. As this research area is advancing rapidly, we encourage community contribution in order to keep this database up to date.