I will talk about joint work with Siegfried Echterhoff and Rufus Willett in which we introduce and study amenable actions of locally compact groups on C*-algebras, building on previous similar notions by Anantharaman-Delaroche for actions of discrete groups. Among the new results we prove an extension of Matsumura's theorem giving a characterisation of the weak containment property (coincidence of full and reduced crossed products) for actions on commutative C*-algebras and give examples showing that this result does not extend to general noncommutative C*-algebras.