The absolute Galois group of the rationals is my primary interest. It contains almost all arithmetic information: eg, solutions to explicit diophantine equations (as in Fermat's Last Theorem). The whole group in general is rather too large an object to study; a better way of understanding the Galois group is through its representations, and this brings out deep connections with other mathematical objects (such as modular forms). For example, given a two dimensional representation of the Galois group satisfying `usual conditions', there should be a modular form whose Fourier coefficients are related to the traces of the representation. The precise correspondences are conjecturally given by the conjectures of Artin (complex representations), Fontaine and Mazur (p-adic representations), and Serre (finite characteristic). It is aspects of these conjectures, both over the rationals and in the setting of totally real number fields, that I am most interested in.