Before the Second International Congress of Mathematicians in 1900, David Hilbert presented 23 problems left to be solved in the upcoming century. The tenth problem asked to devise a process according to which it can be determined whether a Diophantine equation has a solution in the integers. Seventy years later, it was proven that no such general algorithm exists. The reason behind it is that a surprisingly large collection of sets can be expressed as the solution set of a Diophantine equation. For example, there exists a polynomial in 26 variables whose positive values are exactly the prime numbers. In this talk I will go through the proof of H10 and, if time permits, mention some related work of myself. No prior knowledge of logic or number theory is required.