"Quantum correlations are those supporting technologies such as quantum information processing. For realistic applications, one has to consider open quantum systems, that is, in contact with the classical world through lifetime and excitation. Quantum correlations are transferred through emitted photons, electrons, etc. and characterise the quantum structure of the system and its suitability as a quantum device. The state-of-the-art is the Hanbury Brown-Twiss two-photon counting, which is a particular case of the general problem.At the speed of technological progress, it is now becoming possible to measure higher order correlations of quanta characterised in all their attributes. For instance, cross-correlating photons with fixed frequencies and arrival times is now a routine practice in most laboratories worldwide. The correct interpretation and mastering of such techniques will allow a robust implementation of quantum protocols.Theoretically, the computation of such correlations is extremely complicated and tedious as it needs to keep in the calculation all the degrees of freedom for each carrier. I have recently developed a general formalism, called ""the sensing method"", conceptually transparent and improving computations by several orders of magnitude as compared to the previous methods. This allows to deal for the first time with complicated quantum systems, with many degrees of freedom and particles, and to compute Nth-order correlations, with N>2, at arbitrary times and frequencies.The goal of the SQUIRELL project is to develop and disseminate this novel and interdisciplinary theoretical approach in a wide range of quantum systems (cavity QED, superconducting circuits, atomic and semiconductor systems, plasmonic, Bose-Einstein condensates, etc.), by analysing the physics made accessible by the sensing method, by supporting experiments on quantum correlations in a variety of fields and by exploiting correlations to improve and design new quantum devices."