I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations for general linear groups (and general unitary groups). Moreover, our results apply to general reductive groups and have therefore the potential to become widely applicable beyond the case of the general linear group (and general unitary groups).