Low-dimensional quantum magnetism is an important branch of research in the condensed matter physics. Materials that have a low-dimensional magnetic structure often have rich phase diagrams as a consequence of quantum fluctuations and spatial constraints and offer an excellent platform for testing of the theoretical predictions. Quantum transitions between phases, which can be caused by mechanical or magnetic effects, deserve special mention since they are also the subject of a lot of research. In the vicinity of such a transition, quantum fluctuations determine the physical system, and the system behaves universally, independent of the chemical and other specifics of the system itself. As a part of the low-dimensional physics, magnetically isolated clusters like one-dimensional chains such as the Haldane chain, the spin ladders, and two-dimensional planes of which perhaps the most well-known is the group of high-temperature superconductors, have been studied. Nuclear magnetic resonance technique, paired with the Nuclear quadrupole resonance technique, is indispensable in understanding the physics of such materials. These techniques provide direct insight into the magnetic and electric field near the atomic nucleus of interest. The spectroscopic and relaxation measurements of these techniques allow insight into static and dynamic spin susceptibility. Using single crystal samples for the experiment is extremely beneficial, for the NMR technique can resolve angle, and thus provide angle-related information that is lost if the sample is in a powder form. However, to be able to use the single crystal samples, a goniometer must be able to orient the sample in the desired orientation. Thus, in the scope of this thesis, a goniometer was designed and manufactured. This goniometer proved to be very reliable, made to withstand high magnetic fields and to be able to rotate in the low temperature environment. This paper will present the study of two low-dimensional magnetic structures: the spin cluster SeCuO_3 and the Haldane chain m-NO_2PhBNO (BoNO) by nuclear magnetic and quadrupole resonance techniques. Spin clusters possess simple intracluster interactions that can be well described theoretically and confirmed experimentally. SeCuO_3 compound cluster is worth studying because of its elusive simplicity. It is characterized by a wide temperature range of low-dimensional physics, from 8 K when the antiferromagnetic phase arises, to the room temperature at which it still doesn't follow the Curie-Weiss magnetism model. The behavior of a compound over a wide temperature range, from the antiferromagnetic phase to the activation behavior within tetramer spins, will be studied in the scope of this thesis. The Haldane system m-NO2PhBNO is interesting because it is a good candidate for an ideal Haldane chain, given the complete isotropy of the spin. Previous realizations of the Haldane chain of integer spin have a large anisotropy because the spin carrier is an ion of a transition metal group. In contrast, this system is a completely organic compound, where the integer spin is achieved in a molecular electron orbital. In addition, the two critical magnetic fields at which the quantum phase transitions, characteristic of such systems, occur are relatively low, with the upper field at H_c2 ≤ 40 T. We have characterized the phase diagram of the system at low magnetic fields and studied the one-dimensional physics region of the phase diagram described by the Luttinger fluid. In the first chapter of this thesis, Introduction, a brief overview of the magnetism was presented, with special emphasis on low dimensional physics regarding magnetic interactions. Motivation is given for this thesis. In the second chapter, The NMR experiment, overview of the experimental setup is given - the physics of the NMR, electronics and different measurement techniques that were used during the experimental research that is described in this thesis. Design and manufacturing, as well as using the rotator is described in details there. Third and fourth chapters cover experimental research of the tetramer compound SeCuO_3 and Haldane chain system m-NO_2PhBNO, respectively. Structural overview is described at the beginning, followed by theoretical description of expected physics and previous measurements. Main part of these chapters is the presentation of the experimental data generated in the scope of this thesis and its analysis. A vast amount of data was gathered during this doctoral research. The SeCuO_3 was characterized and its orientation determined by the crystallographic X-ray measurements. Phase transition to 3D ordered state was mapped with respect to magnetic field using homemade heat capacity method. Nuclear quadrupolar resonance spectrum was measured in the ordered phase, as well as relaxations. Small magnetic field was used as a perturbation to the quadrupolar Hamiltonian to determine the spin site responsible for the resonance signal. Nuclear magnetic resonance measurements were conducted and line shift was taken. An effort was made to completely understand complicated NMR spectrum, but to no avail at the time of the writing of this thesis. For the m-NO2PhBNO, characterisation and orientation were also done by the crystallographic X-ray. Phase diagram was mapped with homemade heat capacity method at very low temperatures, T < 0.5 K. Nuclear magnetic resonance spectrum was measured, and used to determine the best possible NMR line to conduct relaxation measurements on. Relaxation measurements were conducted at varying magnetic fields and temperatures in the phase region of Luttinger-liquid physics. Fifth chapter is the conclusion and outlook of the entire thesis. Main findings regarding SeCuO_3 are determination of the critical exponent connected to phase transition into anti-ferromagnetic state, lack of the NQR signal for the Cu2 crystal position, determination of the Cu1 crystal position to the visible NQR signal and relaxation and line shift measurements that both provide irrefutable evidence of singlet pairing of the inner tetramer spins, and their activation as the temperature is raised. This activation energy is consistent with spin interaction determined from a non-local methods. For the BoNO compound, a plethora of characterisation methods has been done, including the cooperation with chemistry department and Ruđer Bošković Institute, but also the heat capacity measurement setup made in-house. In the scope of this thesis, Luttinger liquid region of the phase diagram has been studied. It was shown that inside the Luttinger liquid region of the phase diagram the interaction is attractive, as was theoretically predicted for the Haldane chain system. These findings will provide a good basis for the start of measurements at low temperature and high magnetic field facilities, required to fully understand this compound. Additionally, Appendix A contains specifics of the calculations for the eigenvalues and eigenstates for the SeCuO_3 tetramer model.