Abstract: The theory of $\it{enriched}$ categories generalizes naturally the classical theory of categories and has at least as much impact in mathematics as classical category theory. Following ideas of B$\mathrm{\acute{e}}$nabou, Grothendieck, Kelly, Mac Lane, Segal and others, we develop a theory of $\it{weakly}$ $\it{enriched}$ categories; these structures arise naturally when Algebra meets Homotopy and give an alternative approach to $\it{higher}$ $\it{categories}$. In this talk I will give an overview of classical enriched category theory and then will talk about the extension to $\it{weak}$ $\it{enrichment}$.