Additive manufacturing (AM) with concrete, also known as concrete 3D printing, is one of the most interesting approaches for disrupting the construction industry and is currently subject to numerous research activities worldwide. AM has great potential to decrease labour costs and increase the material efficiency and geometric complexity of non-standardised building components. Although prior investigations have shown various fields of application for AM with concrete, the full potential with respect to different structural component types has not been covered yet. With this paper, an up-to-date review of fabrication strategies for the main structural components, (1) walls, (2) columns, (3) slabs, and (4) beams, is provided to identify trends and existing challenges. Therefore, firstly, AM methods and their underlying principles and characteristics for concrete components are presented, and secondly, fabrication strategies for each AM method are shown. The investigation uncovers different AM strategies (direct part vs. indirect “permanent formwork”; in situ, on-site, or off-site), which are currently being used. As a result, future applications of AM will require a hybrid manufacturing strategy combining conventional and additive manufacturing to fully explore its potential.