||Given a system of linear equations over the integers, one can decide if it has an integer solution or not. However, the DPRM theorem gives a negative solution to Hilbert's tenth problem, and therefore there is no algorithm for deciding existence of solutions to Diophantine equations in general; that is, the Diophantine problem of the integers is undecidable. After this, some natural problems appear: Where is the change from decidable to undecidable as we go from linear equations to all Diophantine equations? Are there other rings with interesting arithmetic whose Diophantine problem is undecidable? In this lecture I will not give a complete account of the subject, but instead, an introduction (and invitation) to it.