Quantum phase transitions of the transverse Ising and antiferromagnetic XXZ spin S=1/2 chains are studied using quantum discord. Quantum discord allows the measure of quantum correlations present in many-body quantum systems. It is shown that the amount of quantum correlations increases close to the critical points in opposite to the amount of classical correlations. The observations are in agreement with the information provided by the concurrence which measures the entanglement of the many-body system.