Quantum magnetism remains a hot topic in condensed matter physics due to its complexity and possible powerful and significant applications in data storage and memory. To understand how the materials can achieve these goals, one should have a clear idea about the fundamentals behind it. In this thesis, we focus on three examples that can help us deepen the knowledge in many-body effects, which stand to be crucial for quantum magnetism. (1) The well-known \textbf{CuSO$_4\cdot$5D$_2$O} material has already demonstrated the model behaviour as one-dimensional Heisenberg antiferromagnet in zero and high ($H>H_{sat}$) fields. The fully-polarized magnetic ground state is described by linear spin-wave theory with magnons, whereas at zero field, the excitations are pairs of topological excitations called spinons. In an intermediate field, the dynamic properties are even more complicated. The inelastic spectrum cannot be reproduced without considering exotic elementary excitations and bound states such as psinons and Bethe strings. Although Bethe in 1931 provided an exact solution for 1D Heisenberg systems, there is still no quantitative comparison between theory and experiment. (2) The magnetic ground state and hence the dynamic properties of the gemstone mineral green dioptase, \textbf{Cu$_6[$Si$_6$O$_{18}]\cdot6$H$_2$O} are under debate: starting from controversial theories and continuing with non-explained experimental observations. Dioptase is a quasi-one-dimensional spin chain with dominant antiferromagnetic interactions. Recent studies claim the classical spin chain behaviour and absence of any quantum fluctuations in the system. In contrast, our experimental findings indicate the presence of continuous excitations above and below T$_N$. (3) Newly synthesized material \textbf{CuSb$_2$O$_6$} of rosiaite-type structure tends to become a quantum spin-liquid (QSL) candidate since the magnetic cations Cu$^{2+}$ are arranged in trigonal layers, and no long-range order is observed down to 2~K. The idea of QSL on the triangular lattice was proposed by P. Anderson in 1973. Since his work, a lot of efforts have been made to explore deeper, both theoretically and experimentally, this state. As for now, there are several requirements needed to be met -- small spin number, absence of long-range order or spin freezing, long-range entanglement, and the associated fractional spin excitations. We aim to establish whether or not CuSb$_2$O$_6$ can be considered as a potential quantum spin-liquid candidate employing different techniques suitable for a powder sample.