This study applies numerical and experimental techniques to investigate the effect of process parameters on the density, structure and mechanical properties of pure tungsten specimens fabricated by laser powder bed fusion. A numerical model based on the simplified analysis of a thermal field generated in the powder bed by a moving laser source was used to calculate the melt pool dimensions, predict the density of printed parts and build a cost-effective plan of experiments. Specimens printed using a laser power of 188 W, a scanning speed of 188 mm/s, a hatching space of 80 µm and a layer thickness of 30 µm showed a maximum printed density of 93.2%, an ultimate compression strength of 867 MPa and a maximum strain to failure of ~7.0%, which are in keeping with the standard requirements for tungsten parts obtained using conventional powder metallurgy techniques. Using the optimized printing parameters, selected geometric artifacts were manufactured to characterize the printability limits. A complementary numerical study suggested that decreasing the layer thickness, increasing the laser power, applying hot isostatic pressing and alloying with rhenium are the most promising directions to further improve the physical and mechanical properties of printed tungsten parts.