What is an (\infty,n)-category?

 

I will give an informal introduction to the concept of an $(\infty,n)$-category, discussing how one would recognize an $(\infty,n)$-category, what are (some of) the existing models, and how they compare. In particular, I will discuss how the case of $(\infty,0)$-categories is just the usual theory of spaces, then describe the main models of $(\infty,1)$-categories and how they relate to quasi-categories, and finally mention the work that is being done to model $(\infty,2)$-categories.