The study of the entanglement properties of the ground-state of an extended quantum system has prompted an intense research activity at the crossroad of different disciplines such as statistical mechanics, quantum information, and quantum filed theory. Quantifying the entanglement allowed an elegant and finer characterization of many extended quantum systems. In this talk, after reviewing the results for one-dimensional systems at a quantum critical point that are described by a conformal invariant field theory, I will consider the corrections to the asymptotic behavior. I will show that leading corrections are unusual, in the sense that they depend (in a calculable way) on the entanglement measure and they arise from relevant operators induced by the conical space-time singularities necessary to describe the reduced density matrix.