This thesis is divided into three chapters. In the first chapter we talk about classical computability theory and generalizing the definition of a recursive function to functions with values in \(\mathbb{Z}\) and \(\mathbb{Q}\). In the second chapter we talk about recursively enumerable sets, computable numbers and recursive functions with the codomain \(\mathbb{R}\). We give two characterizations of computable numbers and an example of a non-computable number which is a limit of a recursive sequence. Furthermore, we define recursive functions with values in \(\mathbb{R}\) and generalize results from the first chapter, but also give counterexamples for claims that do not hold by expanding the codomain. That way we show that the set of roots of a recursive function with values in \(\mathbb{R}\) is not necessarily recursive, or even recursively enumerable. In the third chapter we talk about computability in metric spaces. We give the definition and examples of computable metric spaces with results that will serve as a motivation for introducing computability structures on metric spaces. Then we study computability structures and give definitions of separable and maximal computability structures. We prove that every computability structure is a subset of some maximal computability structure and also that every maximal computability structure on Euclidean space is separable. Moreover, we prove that there exists a unique separable computability structure on [0; 1].