The author considers potential theory with respect to a potential kernel \(k=\int^{\infty}_{0}\mu_ tdt\) for a convolution semigroup \((\mu_ t)_{t>0}\) on a (non-Abelian) locally compact group G. It is assumed that there exists a positive Radon measure \(\mu\) on G such that ǩ*\(\mu\) exists and such that ǩ*\(\epsilon {}_ x\blacksquare<\mu\) for all \(x\in G\). Excessive functions and measures are studied, and a Riesz decomposition theorem formulated. The proof presented follows the classical proof of the Abelian case as given in the book of \textit{C. Berg} and \textit{G. Forst} [Potential theory on locally compact Abelian groups (1975; Zbl 0308.31001)]. This, however, does not seem appropriate, since the Deny lemma is unjustified in the non-Abelian case. The Riesz decomposition holds however for standard processes in duality. The Green function is constructed, and it is shown that the convex cone of potentials has the property of proportionality.