In quantum many-body systems interactions between particles create correlations and particles can not be described independently. As a consequence the wave function of the system is a complicated object holding a large amount of information and in most cases it is not possible to obtain it exactly or analytically. Studying correlations in the context of strongly interacting onedimensional (1D) systems is interesting because in this regime we can exactly determine many particle wave function through Fermi-Bose (FB) mapping. FB mapping is in the essence of a Tonks Girardeau (TG) gas proposed in 1960, a theoretical model of 1D Bose gas with strongly repulsive contact interactions between particles. TG gas is experimentally realized in 2004 with rubidium atoms conned in tight atomic wave-guides and it is still a topic of experimental studies. Our goal is the study of correlations in TG gas through the formalism of the single particle density matrix. Single particle density matrix, among other important observable's, contains information about the momentum distribution which can be measured in experiments. We explored correlations in dark soliton-like excitations of the TG gas and found that these correlations give rise to a specific form of the momentum distribution which can be used to experimentally detect these states. In addition to the TG gas, FB mapping was used to construct exact wave functions of 1D Bose-Fermi mixtures (fermions are spin polarized). We derived the formula for the single particle density matrix of the mixture and used it to study correlations and momentum distribution of the mixture in the ground state of double well potential. Due to integrability, these systems are convenient to study stability to perturbations of many-body quantum states in different interaction regimes. We explored stability of non-equilibrium dynamics of 1D interacting Bose gases with respect to small spatial noise potential using the concept of Loschmidt echo (quantum fidelity). We used this concept to explore the pinning quantum phase transition. We found that the ground state fidelity and the Loschmidt echo can be used for diagnostics of the pinning quantum phase transition in experimentally relevant conditions.