Nash problem on spaces of arcs

This thesis is organized as an introduction to the Nash problem on arc spaces and asurvey of all (known to the author) results, which give a positive or negative answer to it.We start with the class of toric and stable toric varieties (Ishii, Kollr, Petrov), includingthe original results of the author. Then we survey the class of normal surface singularities(Plnat, Popescu-Pampu, Reguera, Lejeune-Jalabert, Morales), giving two useful sufficientconditions (Plnat) as well. Finally, a couple of results in three and higher dimensional caseare observed (Plnat, Popescu-Pampu, Perez). In the thesis are presented and discussed alsothree generalizations of the problem (embedded Nash problem, Nash problem for pairs, localNash problem).