This research aims at improving the synergy between 3D printing and topology optimisation. 3D printing, more formally known as additive manufacturing, forms an object by constructing successive layers on top of each other. This procedure of manufacturing offers new possibilities due to the improved design freedom. Topology optimisation is a mathematical design tool that finds the optimal material distribution for a specified problem. Usually the generated optimal designs have a complex shape which are difficult to produce with traditional manufacturing techniques, and post-processing of the design is necessary in order to obtain a manufacturable design. Additive manufacturing makes most of this post-processing redundant due to its unprecedented manufacture abilities. However, despite the greatly improved design freedom, certain geometric restrictions are encountered in popular additive manufacturing techniques like selective laser melting (SLM). Among others, designs need to be self-supporting or overhang-free in order to guarantee a successful production. The best way to deal with this is to incorporate an overhang restriction directly into the topology optimisation, such that optimal designs are generated that also comply with this manufacturing constraint. An overhang constraint is formulated based on the gradient of the density field. The gradient of the density field is used because it is particularly suitable for incorporating the part orientation in the build chamber of the SLM machine into the optimisation. Besides this, it allows for an arbitrary critical angle and is easily extended to 3D. This constraint is implemented in an existing topology optimisation algorithm and tested on numerical examples. Various implementations are tested and compared. Furthermore, a novel method is devised to efficiently include the build orientation into the optimisation. By numerical examples it is shown that this method results in optimal structures and corresponding build directions that are free of overhang.