The well-known Hitchin–Thorpe inequality is an important geometric inequality which states that if a closed, oriented 4-manifold M admits an Einstein metric, then the Euler characteristic \(\chi (M)\) and the signature \(\tau (M)\) of M satisfy \(\displaystyle \begin{aligned} \notag \chi (M)\geq \frac {3}{2} |\tau (M)|. \end{aligned} \) In 1974, N. Hitchin further established a thorough explanation of the equality case. In addition, Hitchin proved that if \((M, g)\) is an Einstein manifold for which equality of this inequality is attained, consequently, either M is flat or its universal cover is a K3 surface, and \((M, g)\) is Ricci-flat. After this important work of Thorpe and Hitchin, there exist many important works related closely to the Hitchin–Thorpe inequality done by many mathematicians. The purpose of this chapter is to delve into recent advancements of the Hitchin–Thorpe inequality and its extensions, showcasing how contemporary studies have extended their applicability.