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The theory of Lé vy processes in Lie groups is not merely an extension of the theory of Lé vy processes in Euclidean spaces. Because of the unique structures possessed by non-commutative Lie groups, these processes exhibit certain interesting limiting properties which are not present for their counterparts in Euclidean spaces. This work provides an introduction to Lé vy processes in general Lie groups, the limiting properties of Lé vy processes in semi-simple Lie groups of non-compact ty and the dynamical behavior of such processes as stochastic flows on certain homogeneous spaces.