The purpose of this seminar is to study the ergodic and entropy theory of the hereditary shift which is introduced by Kerr-Li (Math. Ann. 2007). Hereditary shift generalizes many important subshifts, e.g., bounded density, spacing and beta shifts. The aim of this seminar is to study the topological behavior (e.g., almost specification) and the thermodynamic properties (e.g., the intrinsic ergodicity) of the hereditary or more general shifts. The up-to-date survey of topological and thermodynamic properties of bounded density shifts and the subshifts with the almost specification property is presented herein. Some open questions will be raised in the seminar and hope to solve them together.