In this talk I will give an introduction to three beautifully interconnected topics: scattering amplitudes, polylogarithms, and cluster algebras. I will begin by reviewing recent advances in the calculation and understanding of amplitudes in N=4 Super-Yang-Mills theory. Much of this progress comes from powerful mathematical machinery associated with polylogarithm functions. I will then introduce cluster algebras, a relatively recent area of mathematics with increasing prevalence in physics. Amplitudes in N=4 SYM unexpectedly provide an amazing class of functions that elegantly conjoin ideas from polylogarithms and cluster algebras. As an example, I will describe an algorithm that exploits these connections to calculate two-loop MHV amplitudes for any number of particles.