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During the talk I will review the basic concepts concerning the Majorana fermions and the Majorana modes in the solid state physics. In particular, I will explain the meaning of the Majorana zero modes, address the difference between strong and soft modes as well as the basic concept of quantum computations which utilize the unique properties of the Majorana modes. I will also review the existing experimental support for the Majorana modes and summarize the main challenges for the future experiments. Finally, I will briefly review our research concerning the detection and braiding of the Majorana modes in systems with many-body interactions.