In this thesis, I present the results of my PhD research on the cooling flow problem in galaxy clusters. The centre of relaxed galaxy clusters has a short radiative cooling time suggesting the presence of a massive cooling flow. However, early studies using high resolution X-ray spectroscopy indicated much lower levels of cooling rate. AGN feedback is thought to be the most likely energy source to balance radiative cooling, though the energy transport and dissipation mechanisms are still under debate. In this work, I study whether AGN feedback can actually balance radiative cooling in a large number of galaxy clusters using high resolution X-ray spectroscopy. The first chapter contains the necessary background of the cooling flow problem and AGN feedback. This is followed by a chapter describing data reduction of XMM-Newton observations. In the third chapter, I present a study of 45 nearby cool core galaxy clusters and groups, where I measure the radiative cooling rate in the softest X-ray band. Then I select a small sub sample of bright clusters to understand the mass temperature profile of the gas in Chapter 4. In Chapter 5, I present a deep study of recent XMM-Newton observations of two luminous clusters at intermediate redshift. Finally, I extend my research on 40 more clusters within a much larger range of distances corresponding to redshift up to 0.6.