||Entanglement entropy (EE), which is an important quantity in the quantum information science, has recently attracted a great attention as a new tool for probing the universal properties of quantum many-body systems. In one-dimensional (1D) quantum critical systems, for example, the scaling of EE on a single interval in the system can be used to determine the central charge, a universal number, of the underlying conformal field theory (CFT), with useful applications in numerical simulations. Recent studies have shown that the EE contains more refined information of CFT if it is calculated on two disjoint intervals in the 1D system. Developments are also ongoing for higher-dimensional critical systems. An unusual EE scaling (violating the standard boundary law) in higher-dimensional free fermions has been explained by describing the system as a collection of 1D free fermions. Furthermore, numerical studies have reported that non-Fermi liquids appearing in strongly interacting fermions also show similar unusual EE scalings, providing a useful guidance for constructing metallic states via AdS(anti-de Sitter)/CFT correspondence. Here we review some basic knowledge on EE and these recent developments on the connection between EE and CFT.