Ever since Ax, Kochen [AK65] and Ershov’s [Er65] celebrated results on the model theory of non archimedian local fields, valued fields have been among the “classical” algebraic structures studied in model theory. This is in large part due to the ubiquity of valuations in number theory and algebraic geometry, making the study of valued fields an important bridge between model theory and these subjects and leading to numerous beautiful applications. Most of the subsequent work on the model theory of valued fields focused on the class of henselian valued fields—in particular in the large body of work on (uniform) p-adic, and eventually motivic, integration. However, the earlier seminal work of Robinson [Rob77] focused on the smaller class of the algebraically closed valued fields; and this class eventually came back into the spotlight in the early 2000’s when ideas and techniques imported from geometric stability theory allowed a deep understanding of the structure of algebraically closed valued fields, eventually leading to such masterpieces as the Hrushovski-Kazhdan theory of motivic integration [HK06] and the Hrushovski-Loeser theory of analytic geometry [HL16]. In these notes, our goal is to give a self-contained exposition of two cornerstones of this geometric theory of algebraically closed valued fields. The first is a description of the definable sets in the guise of an elimination of quantifiers, essentially dating back to Robinson’s aforementioned work [Rob77]. The second is a description of all interpretable set, in the guise of the Haskell-Hrushovski-Macpherson elimination of imaginaries [HHM06].