We discuss three di�erent formulations of the equivariant Iwasawa main conjecture
attached to an extension K/k of totally real �elds with Galois group G, where k is a
number �eld and G is a p-adic Lie group of dimension 1 for an odd prime p. All these
formulations are equivalent and hold if Iwasawa's μ-invariant vanishes. Under mild
hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture,
the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture
provided that μ = 0.