  Quantum magnetism in low dimensions has been one of the central areas of
theoretical research for many decades now. One of the key reasons for the long
standing interest in this field has been the existence of simplified models,
which serve as paradigms for understanding the role of strong interactions in
many-electron systems. Although simple, these models quite often can not be
solved exactly. In this review, we discuss a variety of analytical and
numerical methods, which treat the system in a systematic and controlled
manner. The central method employed in all the studies is the density matrix
renormalization group (DMRG) method. This is supported by small scale numerical
exact calculations, and by analytical methods using field theoretic techniques.
We have considered a number of magnetic systems, from magnetic clusters to
extended lattices, and have found some novel quantum ground states and
low-energy elementary excitations. In some cases, we have also employed the
finite-temperature DMRG method to accurately compute the low-temperature
thermodynamic properties such as specific heat and magnetic susceptibility.
