Nanofluid as a special thermal transporting medium has recently received unprecedented attention due to its improved heat transfer performance compared to conventional fluids. Numerous researches have been conducted on the natural convection characteristics of different nanofluids in various configurations of cavities due to the important applications of natural convection in environmental, petrochemical, medical, aviation and space technology, industrial and many more areas. The emergence of a magnetic field as a tool for the manipulation of convective flow and heat transfer behaviours of nanofluids in non-square enclosures has been extensively reviewed. The influence of several variables such as controlling parameters, heat distribution methods, thermal and concentration boundary conditions, magnetic field types, numerical methods, correlation types, nanofluid types, heaters types, numbers and length, and slip conditions, etc., on the magnetohydrodynamic (MHD) natural convection flow and heat transfer behaviours of nanofluid in non-square cavities has been given great attention and brought to the spotlight for discussion. The concepts of bioconvection, micro-polar nanofluid, bio-nanofluid (green nanofluid), ionic nanofluid, and hybrid nanofluid have also been discussed for the first time in relation to natural convection. Special cases of MHD natural convection in non-square cavities involving hybrid nanofluids and micro-polar nanofluids are also presented herein. The application of several numerical methods (which is the major approach studied so far) to investigate the hydromagnetic behaviours of nanofluids in non-square cavities is the focus of this work.