ABSTRACT: A number field K is a finite extension of the rational numbers Q.
 Let K^un be the maximal unramified extension of K inside some algebraic
closure.  The Galois group G_K=Gal(K^un/K) is the 'fundamental group' of
K.  Minkowski proved that G_Q is trivial.  In general, we know very little
about G_K.  Even G_K^ab, which, by class field theory, is the ideal class
group of K, is mysterious (e.g., that G_K^ab is trivial for infinitely
many K is an open problem).  The p-primary component G_K(p) of G_K, has,
however, been studied with some success using a mixture of group theory
and number theory. It is a finitely generated pro-p group, which can be
infinite (Golod-Shafarevich).  I will describe two recent parallel
developments which have shed light on G_K(p): an emerging conjectureal
classfication of finitely generated pro-p groups, and a conjecture of
Fontaine and Mazur characterizing Galois representations arising from
algebraic geometry.