The authors study nonlinear potential theory of an \(L^p\) sub-Markovian semigroup \((T_t:\, t\geq 0)\) on a general measure space \((X,\mu)\). The first question they address is a kernel representation of the semigroup in the form \(T_t u(x)=\int_X u(y)p_t(x,y)\mu(dy)\), and various properties of densities \(p_t(x,y)\). Since the semigroup is not assumed symmetric, it is sometimes necessary to require that the dual semigroup be sub-Markovian. A criterion for this to be true is given. By use of the kernel representation for the semigroup and its \(\Gamma\)-transform \(V_ru:=(1/\Gamma(r/2))\int_0^{\infty} t^{r/2-1}e^{-t} T_tu\, dt\), \(r>0\), \(u\in L^p\), the authors introduce nonlinear potentials, \((r,p)\)-mutual energies and \((r,p)\)-potentials of measures. Several \((r,p)\)-capacities are defined, their (dual) representations given, and properties of equilibrium potentials discussed. Examples are given in the case of \({\mathbb R}^n\) where abstract Bessel potential spaces may be identified with concrete function spaces.