Let \( G \) be a graph. The Lanzhou index, alternatively known as the forgotten coindex, is defined as \( Lz(G)=\sum_{u\in V(G)}\overline{d_{u}}d_{u}^{2}, \) where \( d_{u} \) (resp. \( \overline{d_{u}} \)) represents the degree of vertex \( u \) in \( G \) (resp. \( \overline{G} \)). Research findings substantiate that the Lanzhou index demonstrates enhanced predictive capability compared to both the first Zagreb index and the forgotten index in modeling the logarithmic octanol-water partition coefficient for structural isomers of octane and nonane. This review aims to systematically compiling current extremal results and bounds related to the Lanzhou index. Finally, we outline several open problems as directions for future research.