A conjecture of Fontaine and Mazur says if K is a number field, and S is a finite set of places of K not containing any place of residue characteristic p, then any p-adic linear representation of G_{K,S} has finite image. Furthermore, they conjecture that if S contains all places of residue characteristic p, then any p-adic linear representation of G_{K,S} with infinite image must be wildly ramified. However, by the Golod-Shafarevich criterion, we know that infinite tamely and finitely ramified extensions of K exist. Thus, the Fontaine-Mazur conjectures suggest that the class of p-adic representations is too restrictive for the study of the groups G_{K,S}. In answer to this dilemna, Boston (and others) have suggested studying representations of G_{K,S} on rooted trees. This talk (the first of two) will be an overview of the Fontaine-Mazur conjecture and the Golod-Shafarevich criterion, and will culminate with a discussion of Boston's conjectures about representations on rooted trees. In the next talk, I will summarize a recent manuscript of Aitken, Hajir, and Maire which details a construction that yields natural representations of G_{K,S} on rooted trees. I will describe the construction (which makes use of the iterated mondromy group of a polynomial) and then discuss some open problems and conjectures.