  We give a self-contained introduction to accessible categories and how they
shed light on both model- and set-theoretic questions. We survey for example
recent developments on the study of presentability ranks, a notion of
cardinality localized to a given category, as well as stable independence, a
generalization of pushouts and model-theoretic forking that may interest
mathematicians at large. We give many examples, including recently discovered
connections with homotopy theory and homological algebra. We also discuss
concrete versions of accessible categories (such as abstract elementary
classes), and how they allow nontrivial `element by element' constructions. We
conclude with a new proof of the equivalence between saturated and homogeneous
which does not use the coherence axiom of abstract elementary classes.
