  An edge-colouring of a graph is distinguishing, if the only automorphism
which preserves the colouring is the identity. It has been conjectured that all
but finitely many connected, finite, regular graphs admit a distinguishing
edge-colouring with two colours. We show that all such graphs except $K_2$
admit a distinguishing edge-colouring with three colours. This result also
extends to infinite, locally finite graphs. Furthermore, we are able to show
that there are arbitrary large infinite cardinals $\kappa$ such that every
connected $\kappa$-regular graph has distinguishing edge-colouring with two
colours.
