We study some of the applications of category theory to functional programming, particularly in the context of the Haskell functional programming language, and the Agda dependently typed functional programming language and proof assistant. More specifically, we describe and explain the concepts of category theory needed for conceptualizing and better understanding algebraic data types and folds, functors, monads, and parametrically polymorphic functions. With this purpose, we give a detailed account of categories, functors and endofunctors, natural transformations, monads and Kleisli triples, algebras and initial algebras over endofunctors, among others. In addition, we explore all of these concepts from the standpoints of categories and programming in Haskell, and, in some cases, Agda. In other words, we examine functional programming through category theory.