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The talk is on a joint work with Cornelia Drutu. We introduce a concept of tree-graded metric spaces and use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct (using also ideas of Olshanskii, Erschler and Osin) the first example of a finitely generated group with continuum non-homeomorphic asymptotic cones. Note that by a result of Kramer, Shelah, Tent, and Thomas, continuum is the maximal possible number of different asymptotic cones provided the Continuum Note that by a result of Kramer, Shelah, Tent, and Thomas, continuum is the maximal possible number of different asymptotic cones provided the Continuum Hypothesis is true.