A thesis submitted to the School of Physics, Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, June 2015. / Galaxy clusters are the largest virialised and most recently formed cosmic structures. Their study is therefore a powerful mechanism to probe the large scale properties of the universe. A full understanding of the non-thermal properties of galaxy clusters has not yet been achieved. In particular, the origin of radio halos in galaxy clusters is still a topic of debate today. Several models have been proposed to explain the origin of radio halos but all of these models present some problems either in reproducing observations or in predicting the values of their parameters. Recent observations of radio halos in clusters have shown that in some clusters the morphology of the radio emission appears to be more correlated with the distribution of galaxies than it is with the thermal gas. It has also been observed that while most clusters exhibit a correlation between the radio and X-ray luminosities, there are clear exceptions to this rule. Motivated by these observations, we propose a new theory to explain the origin of radio halos in clusters: that the radio emission observed in clusters is produced by electrons injected by different galaxies and di using in the intra cluster medium. In developing our theory, we study the diffusion of relativistic electrons in galaxies by obtaining a very general solution of the diffusion equation which describes the transport of cosmic rays. This solution allows us to study the spatial, spectral and temporal properties of relativistic electrons and their radio emission for a wide range of cosmic objects. We test this model by applying it to the radio galaxy M51 and reproduce its spatial and spectral properties. We also study the evolution of M51-like galaxies. The model is then applied to study radio halos in clusters of galaxies. We investigate how the properties of these radio halos change when the number and types of sources in a cluster are varied. We compare the results of this model with the observed radio map and density flux spectrum for several galaxy clusters, specifically the low-luminosity cluster CL1446+26 and other clusters with a range of luminosities. Our model provides a very promising explanation of the properties of radio halos in galaxies and clusters of galaxies. The observed properties like the morphology of radio maps in galaxy clusters, the ux spectrum and the correlation between radio and X-ray luminosities, can be reproduced by our model. This is done by considering the combination and evolution of galactic sources having radio properties similar to those of M51. Our model is simple and nevertheless very promising and therefore refining it can lead to a better understanding of the origin of radio halos.