Inspired by the success of graphene, various two-dimensional (2D) non-hexagonal graphene allotropes having sp2-bonded tetragonal rings in free-standing (hypothetical) form and on different substrates have been proposed recently. These systems have also been fabricated after modifying the topology of graphene by chemical processes. In this review, we would like to indicate the role of tetra-rings and the local symmetry breaking on the structural, electronic and optical properties of the graphene system. First-principles computations have demonstrated that the tetragonal graphene (TG) allotrope exhibits appreciable thermodynamic stability. The band structure of the TG nanoribbons (TGNRs) strongly depends on the size and edge geometry. This fact has been supported by the transport properties of TGNRs. The optical properties and Raman modes of this graphene allotrope have been well explored for characterisation purposes. Recently, a tight-binding model was used to unravel the metal-to-semiconductor transition under the influence of external magnetic fluxes. Even the introduction of transition metal atoms into this non-hexagonal network can control the magnetic response of the TG sheet. Furthermore, the collective effect of B-N doping and confinement effect on the structural and electronic properties of TG systems has been investigated. We also suggest future directions to be explored to make the synthesis of T graphene and its various derivatives/allotropes viable for the verification of theoretical predictions. It is observed that these doped systems act as a potential candidate for carbon monoxide gas sensing and current rectification devices. Therefore, all these experimental, numerical and analytical studies related to non-hexagonal TG systems are extremely important from a basic science point of view as well as for applications in sensing, optoelectronic and photonic devices.