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In this talk, we present a necessary and sufficient condition for the existence of recurrent extensions of real self-similar Markov processes. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. To be precise, our main result ensures that a real self-similar Markov process with a finite hitting time of the point zero has a recurrent extension that leaves 0 continuously if and only if the MAP associated, via Lamperti transformation, satisfies the Cramér's condition. In doing so, we solve an old problem originally posed by Lamperti for positive self-similar Markov processes. We generalize Rivero’s (2005, 2007) and Fitzsimmons’s (2006) results to real-valued case. Finally, we describe the recurrent extension of a stable Lévy process.