The correlations of multipartite quantum states can have nonclassical features other than entanglement. After giving a brief overview of the subject, we focus on the issue of establishing a hierarchy between measures of entanglement and compatible measures of general quantum correlations. We analyze a family of measures of general quantum correlations for composite systems, defined in terms of the bipartite entanglement necessarily created between systems and apparatuses during local measurements. For every entanglement monotone E, this operational correspondence provides a different measure Q_E of quantum correlations. Examples of such measures are the relative entropy of quantumness, the quantum deficit, and the negativity of quantumness. In general, we prove that any so defined quantum correlation measure is always greater than (or equal to) the corresponding entanglement between the subsystems, Q_E > E, for arbitrary states of composite quantum systems. In this respect, quantum correlations truly go beyond entanglement. We then conclude with a brief overview of possible applications of quantum correlations without entanglement.