Title: Hilbert's Tenth Problem Today: Main Results and Open Problems

Abstract: Among the 23 problems stated by David Hilbert in 1900 we found: 10. Determination of the Solvability of a Diophantine Equation. 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 by a finite number of operations whether the equation is solvable in rational integers.

The problem was shown to be undecidable in 1970. Since that time over 300 papers were published about simplifications, improvements and applications of this resulting different branches of mathematics. In the lecture I am to survey main achievements in this area and discuss some important related questions that still remain open.