Magnetic Resonance Imaging (MRI) is a non-invasive but slow imaging modality for studying different anatomical and functional aspects of human body. However, it is difficult for a patient to remain motionless during the slow MR acquisition process. The subject motion is one of the main hurdles in MRI due to the fact that the respiratory motion is faster compared to acquisition process resulting in ghosted and blurry recovered images. Cardiac and abdominal MR imaging is mostly affected by respiratory motion. In this thesis, compressive sensing (CS) based new approaches are developed to tackle the respiratory motion in cardiac and abdominal MRI examination. The cost function used by CS based MR recovery algorithms include ?1-norm penalty to exploit the transformed domain sparsity of the acquired MR data. The initial part of dissertation presents a comparison of surrogate functions used to approximate the l1-norm penalty. The experimental work shows that the hyperbolic tangent based function outperforms its competing function in the recovery of static MR images for different acceleration rates and various Gaussian noise levels. Based on these findings, an iterative thresholding algorithm utilizing hyperbolic tangent based ?1-norm approximation is developed to recover free breathing dynamic MR images from sub-sampled k-space data. A block matching algorithm, known as Adaptive Rood Pattern Search (ARPS) is then used to estimate and correct respiratory motion among the recovered images. In the next part, an adaptive thresholding parameter utilizing the MR data statistics is derived and used in wavelet domain shrinkage to recover both static and dynamic MR images. A novel iterative shrinkage thresholding (IST) algorithm based on the derived adaptive parameter is also proposed. Results show that the MR recovery using adaptive threshold is more effective in the presence of motion as compared to fixed threshold value. The final part presents the reduction of motion artifacts in the recovery of under-sampled abdominal and liver dynamic contrast enhanced (DCE) MR images using data binning and low-rank plus sparse (L+S) decomposition. In the data binning, radial k-space data is acquired continuously using golden-angle radial sampling pattern and grouped into various motion states or bins. The respiratory signal for binning is extracted directly from radially acquired k-space data. A compressed sensing (CS)-based L+S matrix decomposition model is then used to reconstruct good quality DCE MR images. The proposed techniques are validated using simulated and clinical MRI data.