All groups under our consideration are finitely generated. Asymptotic cones (AC) of groups were introduced by M.Gromov in 1981. He used them for the description of groups with polynomial growth. AC of groups are homogeneous geodesic, metric spaces. There exists a group having non-homeomorphic cones. All AC of G are R-trees iff the group G is hyperbolic. In a recent joint paper with D.Osin and M.Sapir, we called a group G lacunary hyperbolic (LH) if at least one AC of G is an R-tree. We characterize LH groups as direct limits of hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. We show that the class of LH groups is very large. Many group-theoretical couner-examples (E.G., some Tarski monsters) are LH groups. Among new examples, we construct a group having an AC with a non-trivial countable fundamental group. This solves Gromov's problem of 1993.