One of the most important issues in data stream processing systems is to use operator migration to handle highly variable workloads in a cost-efficient manner and adapt to the needs at any given time on demand. Operator migration is a complex process that involves changes in the state and stream management of a running query, typically without any loss of data, and with as little disruption to the execution as possible. This survey provides an overview of solutions for operator migration from a historical perspective as well as the perspective of the goal of migration. It introduces a conceptual model of operator migration to establish a unified terminology and classify existing solutions. Existing work in the area is analyzed to separate the mechanism of migration from the decision to migrate the data. In case of the latter, a cost-benefit analysis is emphasized that is important for operator migration but is often only implicitly addressed, or is neglected altogether. A description of the available solutions provides the reader with a good understanding of the design alternatives from an algorithmic viewpoint. We complement this with an empirical study to provide quantitative insights on the impact of different design alternatives on the mechanisms of migration.