Natural cellular structures inspire numerous engineering applications due to their lightweight and good load-bearing capabilities. Variations in their structural configurations result in diverse mechanical properties, rendering them ideal for bioinspired designs. This review explores the occurrence, properties, and applications of cellular structures in nature and their engineering counterparts built via additive manufacturing (AM). Additive manufacturing technologies enable replication of natural cellular structures with enhanced geometrical complexity. However, they face challenges associated with scale-based geometrical constraints and minimum printable size of various features. Current constraints and techniques for overcoming these challenges, including advancements in nanotechnology, multiscale modelling, novel biomimetic designs, and improved mechanical testing methods are discussed in this paper. It is noted in the paper that it becomes increasingly possible with these advancements to optimise bioinspired engineering parts for complicated applications.