  Quantum information theory is built upon the realisation that quantum
resources like coherence and entanglement can be exploited for novel or
enhanced ways of transmitting and manipulating information, such as quantum
cryptography, teleportation, and quantum computing. We now know that there is
potentially much more than entanglement behind the power of quantum information
processing. There exist more general forms of non-classical correlations,
stemming from fundamental principles such as the necessary disturbance induced
by a local measurement, or the persistence of quantum coherence in all possible
local bases. These signatures can be identified and are resilient in almost all
quantum states, and have been linked to the enhanced performance of certain
quantum protocols over classical ones in noisy conditions. Their presence
represents, among other things, one of the most essential manifestations of
quantumness in cooperative systems, from the subatomic to the macroscopic
domain. In this work we give an overview of the current quest for a proper
understanding and characterisation of the frontier between classical and
quantum correlations in composite states. We focus on various approaches to
define and quantify general quantum correlations, based on different yet
interlinked physical perspectives, and comment on the operational significance
of the ensuing measures for quantum technology tasks such as information
encoding, distribution, discrimination and metrology. We then provide a broader
outlook of a few applications in which quantumness beyond entanglement looks
fit to play a key role.
