While F-theory models readily admit relatively simple representations, it is difficult to construct models with “exotic” representations that go beyond these simple types. This talk will discuss ways of systematically constructing and understanding Weierstrass models with these exotic representations. For non-abelian groups, I will focus on “higher genus” representations, which involve 7-branes wrapping singular divisors. While models with higher genus representations involve intricate, complicated structures, they can be systematically constructed using techniques related to the normalization of singular varieties. I will also describe some results regarding non-abelian representations and matter spectra that cannot be realized in F-theory compactifications. I will then turn to the issue of matter in U(1) models with large charges. First, I will discuss a new strategy for constructing models with charge-3 matter that has interesting parallels with the techniques for constructing higher genus representations. I will also describe the construction of explicit models with charge-4 matter. The talk will conclude with some conjectures on matter with charges larger than 4.