Lattice structures are useful in the aerospace, automotive, infrastructural, and medical fields due to the way they incorporate a lightweight design and good mechanical properties, because of their hollow shapes. This review paper documents work carried out using various analytical models for lattice structures designed with different polygon hollow shapes, for loading in the in-plane and out-of-plane directions, in order to advise their ranking in terms of mechanical behaviour. A primer on lattice structures and polygon hollow shapes is first provided. This is followed by a review of relevant analytical models applied to lattice structures with various polygon hollow shapes that are available in the literature, and then a ranking of the polygon hollow structures in terms of their mechanical properties is performed. Following on from this, a review of the mechanical properties of polygon hollow structures is given. Engineering applications of different polygon hollow structures are then identified. A next-generation structural optimisation and design guide is then highlighted, and some of the primary prospective areas to be focused on when designing lattice parts are pointed out. The last section highlights current challenges, as well as recommendations for extending the use of design for the additive manufacturing of lattice parts.