We present an up-to-date survey on the topic of adaptive mathematical morphology. A broad review of research performed within the field is provided, as well as an in-depth summary of the theoretical advances within the field. Adaptivity can come in many different ways, based on different attributes, measures, and parameters. Similarities and differences between a few selected methods for adaptive structuring elements are considered, providing perspective on the consequences of different types of adaptivity. We also provide a brief analysis of perspectives and trends within the field, discussing possible directions for future studies.