A recent trend in algebraic topology is an increased focus on various notions of "higher categories". These are structures which add higher dimensional data to the objects and composable arrows between them that make up an ordinary category. The development of the theory of higher categories has been largely motivated by particular examples where higher dimensional information plays an important role. I will build up the notion of a weak $\infty$-category from scratch (no background on categories assumed) through examples coming from topological spaces, show how these structures arise naturally in Homotopy Type Theory and manifold theory, and time permitting discuss how alternative definitions of higher categories can be more convenient in practice.