  We demonstrate how to create artificial external non-Abelian gauge potentials
acting on cold atoms in optical lattices. The method employs $n$ internal
states of atoms and laser assisted state sensitive tunneling. Thus, dynamics
are communicated by unitary $n\times n$-matrices. By experimental control of
the tunneling parameters, the system can be made truly non-Abelian. We show
that single particle dynamics in the case of intense U(2) vector potentials
lead to a generalized Hofstadter butterfly spectrum which shows a complex
``moth''-like structure. We discuss the possibility to employ non-Abelian
interferometry (Aharonov-Bohm effect) and address methods to realize matter
dynamics in specific classes of lattice gauge fields.
