Motivated by the famous Mazur-Ulam theorem in this paper we study algebraic properties of isometries between metric groups. We present some general results on so-called d-preserving maps between subsets of groups and apply them in several directions. We consider d-preserving maps on certain groups of continuous functions defined on compact Hausdorff spaces and describe the structure of isometries between groups of functions mapping into the circle group T. Finally, we show a generalization of the Mazur-Ulam theorem for commutative groups and present a metric characterization of normed real-linear spaces among commutative metric groups.