Abstract. The Sasaki distance between a pair of bound vectors AB and CD in a Euclidean space is (dist(A,C)^2+|AD'-AB|^2)^(1/2), where AD' is the result of the parallel translation of the vector CD to the point A. Hence, the tangent bundle of a Euclidean space is the space of bound vectors furnished with the Sasaki distance. The construction of the Sasaki distance in a Riemannian space is based on the Levi-Civita parallelism. We present a substitute for the tangent bundle T(M) of a Riemannian manifold M endowed with the Sasaki distance in a metric space. As one expects, M can be isometrically imbedded into T(M), and for a Riemannian space with at least twice continuously differentiable metric tensor our construction produces the standard tangent bundle and the standard Sasaki distance on it. Our construction makes sense for any metric space, in particular for non manifolds. As an example, we present a tangent "bundle" of a graph. If time permits, we will indicate the connection of the construction of the generalized tangent bundle and a solution of a long-standing problem on a metric characterization of Riemannian spaces.