This thesis focuses on two aspects of Quantum Information Processing (QIP) and contains experimental implementation by Nuclear Magnetic Resonance (NMR) spectroscopy. The two aspects are: (i) development of novel methodologies for improved or fault tolerant QIP using longer lived states and geometric phases and (ii) implementation of certain quantum algorithms and theorems by NMR. In the first chapter a general introduction to Quantum Information Processing and its implementation using NMR as well as a description of NMR Hamiltonians and NMR relaxation using Redfield theory and magnetization modes are given. The second chapter contains a study of relaxation of Pseudo Pure States (PPS). PPS are specially prepared initial states from where computation begins. These states, being non-equilibrium states, relax with time and hence introduce error in computation. In this chapter we have studied the role of Cross-Correlations in relaxation of PPS. The third and fourth chapters, respectively report observation of cyclic and non-cyclic geometric phases. When the state of a qubit is subjected to evolution either adiabatically or non-adiabatically along the surface of the Bloch sphere, the qubit sometimes gain a phase factor apart from the dynamic phase. This is known as the Geometric phase, as it depends only on the geometry of the path of evolution. Geometric phase is used in Fault tolerant QIP. In these two chapters we have demonstrated how geometric phases of a qubit can be measured using NMR. The fifth and sixth chapters contain the implementations of “No Deletion” and “No Cloning” (quantum triplicator for partially known states) theorems. No Cloning and No Deletion theorems are closely related. The former states that an unknown quantum states can not be copied perfectly while the later states that an unknown state can not be deleted perfectly either. In these two chapters we have discussed about experimental implementation of the two theorems. The last chapter contains implementation of “Deutsch-Jozsa” algorithm in strongly dipolar coupled spin systems. Dipolar couplings being larger than the scalar couplings provide better opportunity for scaling up to larger number of qubits. However, strongly coupled systems offer few experimental challenges as well. This chapter demonstrates how a strongly coupled system can be used in NMR QIP.