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The study of \emph{quantum magnetism} remains at the forefront of condensed matter physics. Spin models provide a large class of many-body systems, in which cooperative quantum phenomena can be studied in a controlled way. In addition, the \emph{neutron scattering} technique offers a near-ideal tool to probe the state of magnetic materials. This thesis presents neutron scattering studies of three selected materials, each representing an important aspect of quantum magnetism. \begin{itemize} \item CuGeO$_3$ is a quasi one-dimensional $S=1/2$ spin--Peierls material. This is an example of a system that has a \emph{quantum ground state} with no classical analogue. The spins dimerize to form a coherent non-magnetic singlet, where the expectation value of each individual spin is zero, as if they were `hidden'. As a consequence, the \emph{excitations} (called solitons) are different from the spin waves of a classical system. In a high magnetic field, the solitons can be condensed to form a periodic lattice. Through neutron scattering measurements, the structure of this soliton lattice has been determined, and the excitations in the soliton phase have been identified. \item Cu(DCOO)$_2 \cdot$4D$_2$O is a two-dimensional $S=1/2$ Heisenberg antiferromagnet on a square lattice. The $T=0$ ground state of this system has long range order similar to the classical system. But the order parameter is reduced by \emph{quantum fluctuations}, and the physical observables are \emph{renormalized}. In particular, it was found that the spin wave dispersion is non-uniformly renormalized. At finite temperatures long range order is destroyed by thermal and quantum fluctuations, which act together. Still, there are strong \emph{correlations}, which on short length and time scales resembles the long range order. The temperature dependence of the correlation length and the excitation spectrum has been measured using two specialized neutron scattering methods. \item LiHoF$_4$ is a three- dimensional Ising ferromagnet, in which the magnetic order can be destroyed even at $T=0$ by applying a large magnetic field transverse to the Ising axis. Ordinary phase transitions occur as a function of temperature, when the thermal fluctuations become strong enough to destroy the order. At $T=0$ there are no thermal fluctuations and the transition is driven by quantum fluctuations, which are controlled by some external parameter,in this case the magnetic field. It is important to understand the universal behaviour of such \emph{quantum phase transitions}, as several novel phenomena in solid state physics may be related to the proximity of a \emph{quantum critical point}. Using inelastic neutron scattering the behaviour of the excitations around the quantum critical point in LiHoF$_4$ has been investigated. \end{itemize} This thesis was submitted in partial fulfillment of the requirements for a Ph.D.\ in physics at the University of Copenhagen. The work presented was carried out in the Department of Condensed Matter Physics and Chemistry at Ris\o{} National Laboratory, Denmark. The supervisors were Des McMowrrow from Ris\o{} and Jens Jensen from the University of Copenhagen.