  Quantum correlations can be used as a resource for quantum computing, eg for
quantum state manipulation, and for quantum sensing, eg for creating
non-classical states which allow to achieve the quantum advantage regime. This
review collects the predictions coming from a family of exact solutions which
allows to describe the many-body wavefunction of strongly correlated quantum
fluids confined by a tight waveguide and subjected to any form of longitudinal
confinement. It directly describes the experiments with trapped ultracold atoms
where the strongly correlated regime in one dimension has been achieved. The
exact solution applies to bosons, fermions and mixtures. It allows to obtain
experimental observables such as the density profiles and momentum distribution
at all momentum scales, beyond the Luttinger liquid approach. It also predicts
the exact quantum dynamics at all the times, including the small oscillations
regime yielding the collective modes of the system and the large quench regime
where the system parameters are changed considerably. The solution can be
extended to describe finite-temperature conditions, spin and magnetization
effects. The review illustrates the idea of the solution, presents the key
theoretical achievements and the main experiments on strongly correlated
one-dimensional quantum gases.
