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Following original idea1 leading to a discovery of Topological Insulatorswe describe the recent developments of the subject in a detail.In particular we focus on TopologicalSuperconductors and Majorana Fermions. SuchMajoranashave strong potential to be used invarious graphene devices2 as well as infuture topological adiabatic quantum computers3 due to their non-Abelianbraiding statistics. Wedescribe the theory of topological insulators and superconductor and show how Majorana fermions and topological superconductivitymay arisethere considering thespinlesspx±ipy superconductors and hybrid systems.Graphene is not flat and has microscopic latticenano-corrugations inherent to all two-dimensional crystals4. We showthat such corrugations may provide channeling opportunitiesfor electrons5 that can be used in a new design of p-n junctions and transistors2 or for a creation of Majoranas. The graphene latticedistortions can not only generate the state of topological insulator but also induce the magnetization oscillations and the Hofstadterbutterfly in graphene flakes6. We discuss also physical properties of Zenertunnelling nano-devices7 and Aharonov-Bohm effect ingraphene nanoring focusing on the case when there are arising levitons8. Graphene bubbles is another example where topologicalstates may exist9. We also discuss tunneling, stochastic and extraordinary magnetoresistance phenomena arising in these systems10,11,12.