1 Tangent spaces to metric spaces Oleksiy Dovgoshey Institute of Applied Mathematics and Mechanics of NASU, R.Luxemburg str. 74, Donetsk 83114, Ukraine; dovgoshey@iamm.ac.donetsk.ua Olli Martio Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FI-00014 University of Helsinki, Finland; olli.martio@helsinki.fi Abstract. We introduce a tangent space at an arbitrary point of a general metric space. It is proved that all tangent spaces are complete. The conditions under which these spaces have a finite cardinality are found. Key words: Metric spaces; Tangent spaces. 2000 Mathematics Subject Classification: 54E35. 1 Introduction. Main definitions. The recent achievements in the metric space theory are closely related to some gener- alizations of differentiation. The concept of upper gradient [HeKo] and [Sh], Cheeger’s notion of differentiability for Rademacher’s theorem in certain metric measure spaces [Ch], the metric derivative in the studies of metric space valued functions of bounded varia- tion [Am], [AmTi] and the Lipshitz type approach in [Ha] are interesting and important examples of such generalizations. These generalizations of the differentiability usually lead to nontrivial results only for assumption that metric spaces have ”sufficiently many” rectifiable curves. The our main goal is the introduction of the notion of ”differentiable” functions from a metric space X to a metric space Y for arbitrary X and Y . We define ”tangent” spaces at a point of a metric space as some quotient space of the sequences which converge to this point and after that introduce the ”derivatives” of functions as corresponding quotient maps. Let (X, d) be a metric space and let a be point of X . Fix a sequence ˜ r of positive real