What is an $\infty$-category, and why are algebraic topologists so obsessed with them? In this infosession I will introduce $\infty$-categories, starting from the central notion of a homotopy. We will take the point of view that all groups ought to be symmetry groups, and explore the implications of this for the fundamental group and higher homotopy groups of a topological space. If time permits, I will say something brief about how simplicial sets and Jacob Lurie’s cult of triangles fit into all this.