The Scott rank of a structure is a measure of its complexity. I will talk about a few results about the Scott ranks of computable structures, focusing on structures of high Scott rank and in particular of Scott rank $\omega_1^{CK}$. We will start with a few different constructions of structures of this Scott rank with different properties, such as one whose infinitary theory is not countably categorical. Then we will argue that there is no natural/canonical construction of a computable structure of Scott rank $\omega_1^{CK}$.