Hilbert's tenth problem asks for a ``process'' to decide if a Diophantine equation has integer solutions. This turned out to be impossible by the works of Davis--Putnam--Robinson and Matiyasevich. The talk will start with a basic introduction to computation theory, formalizing the notion of ``process''. It will then sketch the main ideas of the negative solution to Hilbert's tenth problem, which shows the much stronger conclusion that in some sense, Diophantine equations can simulate all computations.