I will talk about the following problem: which systems of Diophantine equations have the property that, in every finite colouring of the natural numbers, there is always a monochromatic solution of the system? For example it is well-known that the equation $z=x+y$ has this property, known as "partition regularity", and that so does the equation $t=xy$. In contrast, whether the system $x+y-z = 0 = xy-t$ is partition regular is one of the most long-standing open problems in the area. Partition regularity of systems of Diophantine equations is one of those problems creating fertile ground for interaction between different areas of mathematics. In this case, the list includes ergodic theory, discrete harmonic analysis, topological dynamics, algebra in the space of ultrafilters, nonstandard analysis and model theory. I will give an introduction, aimed at non-specialists, to the nonstandard-analytic/model-theoretic approach to this problem. I will then present some recent result of Mauro Di Nasso, Lorenzo Luperi Baglini, Mariaclara Ragosta, Alessandro Vegnuti and myself about the partition regularity of certain systems of quadratic equations. In particular, we show that $x+y-z = 0 = tx-y$ is partition regular.