Abstract: The usual source of Galois representations is geometry. If V is a variety over Q (for example), we can extend scalars to \bar Q (the field of algebraic numbers), and let Gal(\bar Q/Q) act on the cohomology groups of V.

I intend to briefly review this construction and then to describe a quite different class of Galois representations which are constructed by arithmetic and group theory rather than by geometry. It turns out that these representations are much less rigid then the usual kind.

Time permitting, I will say something about some pathological examples Khare, Ramakrishna, and I have constructed.