Laser Powder Bed Fusion is a leading additive manufacturing technology, which has been used successfully with a range of lower melting point materials (titanium alloys, nickel alloys, steels). This work looks to extend its use to refractory metals, such as those considered in this paper where the behaviour of pure tungsten powder is investigated. A strategy for fabricating high density parts was developed by creating a process map in which the effect of laser energy density was studied. The process quality was assessed using different techniques including light optical microscopy, XCT, SEM and EBSD. The results showed that the laser energy density was adequate to process tungsten to produce functional parts. The bulk density and optically determined densities, under different process conditions, ranged from 94 to 98%, but there was evidence of micro cracks and defects in specimens due to micro- and macro-scale residual stress. Analysis of the microstructure and local crystallographic texture showed that the melt pool formed under the laser beam favoured solidification in a preferred orientation by an epitaxial growth mechanism. The EBSD local texture analysis of the tungsten specimens showed a <111>//Z preferential fibre texture, parallel to the build direction.