Abstract: In 1900, Hilbert posed a famous list of 23 problems, which shaped the mathematics of the following century. We shall talk about Hilber tenth problem: “Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.” The problem can be modernized in the following way: Write a program which given as parameter a polynomial P(X1,…Xn) in multiple variable, with integers coefficient, find whether or not P(x1,…xn)=0 has a solution, where x1…xn are integers. A lot of work on the subject was done by Martin Davis, Julia Robinson, and Hilary Putnam. Finally Yuri Matiyasevich, in 1970, did the last step, proving that there is no solution to Hilbert tenth problem. To be more precise there is no programs that can work for all polynomials. The goal of this talk is to discuss the result and various related problems: Is a similar problem solvable if we want to find a root in the natural numbers? in the rational numbers? or in the real numbers?