Diophantine geometry is the study of integral solutions to polynomial equations. For instance, for integers a,b,c >1 satisfying 1/a + 1/b + 1/c < 1,              Darmon and Granville proved that the individual generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions. Conjecturally something stronger is true: for all integegers a,b,c > 2 there are no non-trivial solutions.

I'll discuss various Diophantine problems, with an emphasis on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and recent partial progress toward the uniformity conjecture.