Shear and bulk viscosities of deconfined gluonic matter are investigated within an effective kinetic theory by describing the strongly interacting medium phenomenologically in terms of quasiparticle excitations with medium-dependent self-energies. We show that the resulting transport coefficients reproduce the parametric dependencies on temperature and coupling obtained in perturbative QCD at large temperatures and small running coupling. The extrapolation into the nonperturbative regime results in a decreasing specific shear viscosity with decreasing temperature, exhibiting a minimum in the vicinity of the deconfinement transition, while the specific bulk viscosity is sizable in this region, falling off rapidly with increasing temperature. The temperature dependence of specific shear and bulk viscosities found within this quasiparticle description of the pure gluon plasma is in agreement with available lattice QCD results.Shear and bulk viscosities of deconfined gluonic matter are investigated within an effective kinetic theory by describing the strongly interacting medium phenomenologically in terms of quasiparticle excitations with medium-dependent self-energies. We show that the resulting transport coefficients reproduce the parametric dependencies on temperature and coupling obtained in perturbative QCD at large temperatures and small running coupling. The extrapolation into the nonperturbative regime results in a decreasing specific shear viscosity with decreasing temperature, exhibiting a minimum in the vicinity of the deconfinement transition, while the specific bulk viscosity is sizable in this region, falling off rapidly with increasing temperature. The temperature dependence of specific shear and bulk viscosities found within this quasiparticle description of the pure gluon plasma is in agreement with available lattice QCD results.