Counting bitangents of quartic curves --- arithmetic, real, tropical

Already Plücker knew that a smooth complex plane quartic curve has exactly 28 bitangents. Bitangents of quartic curves are related to a variety of mathematical problems. They appear in one of Arnold's trinities, together with lines in a cubic surface and 120 tritangent planes of a sextic space curve. Together with his group, Bernd Sturmfels has revived an interest in these topics and provided computational approaches and new constructions. In this talk, we review known results about counts of bitangents under variation of the ground field. Special focus will be on counting in the tropical world, and its relations to real and arithmetic counts. We end with new results concerning the arithmetic multiplicity of tropical bitangent classes, based on joint work in progress with Sam Payne and Kris Shaw.