The edge-spectral-Turán type problem is also called the Brualdi-Hoffman-Turán type problem, which is a central topic in spectral graph theory, seeking to determine the maximum spectral radius $λ(G)$ of an $F$-free graph $G$ with $m$ edges. This problem has attracted significant attention in recent years. In this paper, we will sort out several closely related results in this type of problem and then propose some conjectures for further research.