Below their bulk melting temperature, polycrystalline materials may exhibit the premelting phenomenon, with the grain boundaries (GB) turning into liquidlike nanofilms. This structural change has several important consequences, for example in terms of mechanical properties. We study using phase field simulations the grain evolution in the premelted polycrystal of a pure substance. The dynamics is governed by heat fluxes across the liquid films, across the liquid pockets at the triple junctions, and within the grains. Depending on the dimensionless undercooling Δ and the grain size λ, we identify different regimes. When Δ≪1, the width ρ∼δ/Δ of the liquid pockets is much larger than the atomic distance δ and the dynamics becomes independent of the free energy of the dry GBs. Then, when λ≫ρ, the premelted GBs evolve according to a curvature-driven dynamics, while the grain evolution corresponds to a usual Ostwald coarsening when the liquid pockets overlap, i.e. when λ∼ρ. Within the curvature-driven regime, the triple junctions maintain their equilibrium configuration with 2π/3 angles when λ≫ρ/Δ, while a non-equilibrium configuration develops when λ∼ρ/Δ. We analyze and confirm these effects through two-dimensional phase field simulations. We finally relate our findings to experiments on ceramics exhibiting an anisotropic abnormal grain growth. Supported by additional phase field simulations, we propose a scenario for this phenomenon based on the competition between the velocity of the quasi-liquid layer propagating along the dry GBs and the curvature-driven velocity of the already pre-melted GBs.