In this thesis we study computable metric spaces, with special focus on semicomputable sets in computable metric spaces. First we study the theory of recursive functions from \(\N^k\) to \(\N^m\) and recursively enunmerable sets in \(\N^k\). Also, we extend the notion of recursive functions to functions from \(\N^k\) to \(\Z, \Q\) and \(\R\). We define the concepts of computable points, computable sequences and computable sets in \(\R\). We also study recursive recursively bounded functions, which we use in proofs in the second chapter. In the second chapter we introduce the notion of a computable metric space and extend the notions of computable points, sequences and sets to abstract computable metric spaces. We also define countably enumerable and semicomputable sets in computable metric spaces. Those are two different generalizations of computable sets. We prove some results about those sets that we will need. At the end, we prove the main result, that a semicomputable arc with computable endpoints is necessarily a computable set, and a similar result for the chainable continua.