Simplicity theory, a core line of research in pure model theory, is built upon a tight connection between a combinatorial dividing line (not having the tree property) and a theory of independence (non-forking independence).  This notion of independence, which generalizes linear independence in vector spaces and algebraic independence in algebraically closed fields, is a key tool in the model-theoretic analysis of concrete mathematical structures.  In work of Chatzidakis and work of Granger, related notions of independence were constructed by ad hoc algebraic means for new examples with non-simple theories.  In order to understand these constructions, we introduced Kim-independence which enjoys a tight connection to the dividing line NSOP_1 and explains the work of Chatzidakis and of Granger on the basis of a general theory.  We will survey this work and discuss recent applications