Let G be a reductive group. Following Gross, and generalizing Serre's classical notion in the two-dimensional case, I will define what it means for a G-valued representation of the Galois group of a (totally real) number field to be odd. This notion provides a natural setting for studying generalizations of the familiar properties of odd two-dimensional Galois representations. I will then describe recent joint work with N.

Fakhruddin and C. Khare on the existence of geometric lifts of odd, G-irreducible representations, with an emphasis on novel examples for both classical and exceptional groups.