This was a talk I gave at the Set Theory Workshop at the Mathematisches Forschungsinstitut in Oberwolfach, Germany, 12-17 January 2025.

Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable first-order language there is a structure A of size less than κ, such that B is covered by elementary images of A in B. Is there any such cardinal? Is the principle consistent? Does it have large cardinal strength? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.