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The strength of polycrystals is largely controlled by the interaction between lattice dislocations and grain boundaries. The atomistic details of these interactions are difficult to discern even by advanced high-resolution microscopy methods. In this paper we present results of atomistic simulations of interactions between an edge dislocation and three symmetric tilt grain boundaries in body-centred cubic tungsten. Our simulations reveal that the outcome of the dislocation-grain-boundary interaction depends sensitively on the grain boundary structure, the geometry of the slip systems in neighbouring grains, and the precise location of the interaction within the grain boundary. A detailed analysis of the evolution of the grain boundary structures and local stress fields during dislocation absorption and transmission is provided.