The jet schemes and the arc scheme of an algebraic variety give important information about its singularities, in particular about their invariants. A major problem in the area was proposed by J. Nash in 1968. The main goal of this article is to review the results obtained about Nash problem, and some of its modifications. The first three chapters contain an introduction to the subject, including the Nash theorem and a counter-example in dimension 4. The remaining chapters contain results about toric and stable toric varieties, algebraic surfaces, and the higher dimensional case. Also, the embedded Nash problem, the Nash problem for pairs and the local Nash problem are introduced, with some results discussed.

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I graduated from Sofia University, Bulgaria. In 2001 I became a Ph.D. student at the University of Georgia, GA, working in arc spaces, motivic integration and toric varieties, with thesis advisor V. Alexeev. In 2007 I gained a position at Purdue University, IN. Last academic year I spent as a postdoc at ENS, Paris.