The topic of Diophantine equations is an area of mathematics devoted to studying integer solutions of polynomial equations. They have been around since the ancient Greeks, but as we have barely scratched the surface of their rich structure, they will no doubt continue to draw our attention for many years to come. Part of their appeal lies in how they provide a bridge between very different fields, sometimes resisting attack unless more than one technique is brought into play. In this talk I will give a glimpse of the subject, focusing on questions such as: Can we find solutions? Can we count them? Can we explain when solutions go missing?