What is an (∞, n)-category?

Several algebraic structures of interest that we learn and study in math, such as groups or vector spaces, consist of endowing a set with one or more operations subject to axioms. There are many interesting phenomena in mathematical physics, algebraic geometry, and type theory that seem formalizable by one of such algebraic structures at first glance, although with a closer look the axioms are only met up to a "higher isomorphism". In order to accommodate those situations, one needs to acknowledge the presence of higher structures, and replace the role played by the usual equality relation with the relation induced by higher isomorphisms, which leads to the notion of an (∞,n)-category. In this talk we will describe a couple of different situations across math where the presence of higher structures is observable, discuss the complications that arise, and give an idea of the different approaches that can be taken to work with higher categories rather than ordinary categories.