Observations at radio wavelengths demonstrate the existence of cosmic rays and magnetic fields in the Universe. Studies of galaxy clusters have revealed sources of diffuse radio emission associated with the merger driven shocks and turbulence in the ICM: relics and halos. This thesis present the results obtained from deep radio observations of two individual galaxy clusters. The galaxy cluster 1RXS J0603.3+4214 hosts a bright relic, known as the Toothbrush, and a giant radio halo. The cluster was observed with the VLA covering a frequency range of 1 2GHz. The new VLA images provide an unprecedented view of the Toothbrush, revealing enigmatic filamentary structures. The complexity of the filamentary structures rule out the fact that relics are caused by a smooth shock surface. In L-band, the handle of the Toothbrush is strongly polarized, as high as 60%, while the brush is almost completely depolarized. The fractional polarization in the handle decreases only moderately towards longer wavelength. Rotation Measure (RM) synthesis analysis reveals that the filamentary features in the low density region (B3) show a shift in RM of 30 radm 2 while in the denser region (B2), the shift in RM increases to 50 radm 2. The VLA observations confirm the presence of extended halo. The average spectral index of the halo is 1.16 ± 0.05. The southern part of the halo is steeper and is possibly related to a shock. Excluding the southernmost part, the halo morphology is strikingly similar to the X-ray morphology. The sensitive high resolution radio maps also reveal thirty-two previously undetected compact sources within the halo region. For another cluster CIZAJ0649.3+1801, we confirm the presence of a diffuse emission source. The cluster was observed with the WSRT. The source is polarized, has a steep spectrum, and shows a hint of spectral gradient towards the cluster center. This evidence suggests that it is a radio relic.