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Lava domes and lava flows are major manifestations of effusive volcanic eruptions. Less viscous lava tends to flow long distances, depending on the volcanic slope topography, the eruption rate, and the viscosity of the erupted magma. When magma is highly viscous, its eruption to the surface leads to the formation of lava domes and their growth. The meshless smoothed particle hydrodynamics (SPH) method is used in this paper to simulate lava dynamics. We describe the SPH method and present a numerical algorithm to compute lava dynamics models. The numerical method is verified by solving a model of cylindrical dam-break fluid flow, and the modelled results are compared to the analytical solution of the axisymmetric thin-layer viscous current problem. The SPH method is applied to study three models of lava advancement along the volcanic slope, when the lava viscosity is constant, depends on time and on the volume fraction of crystals in the lava. Simulation results show characteristic features of lava flows, such as lava channel and tube formation, and lava domes, such as the formation of a highly viscous carapace versus a less viscous dome core. Finally, the simulation results and their dependence on a particle size in the SPH method are discussed.