André Joyal's theory of quasi-categories gives a setting into which both homotopy theory of spaces (as Kan complexes) and the theory of categories (as their simplicial nerves) both embed. As such it is a natural setting for weakly associative or weakly commutative categorified structures--in particular, Lurie uses this theory to build his "Derived Algebraic Geometry". I will give an introduction to Joyal's quasi-categories intended for an audience which is not expert in simplicial homotopy theory, model theory, or higher category theory. --