In MR imaging, techniques for acquisition of reduced data (Rapid MR imaging) are being explored to obtain high-quality images to satisfy the conflicting requirements of simultaneous high spatial and temporal resolution, required for functional studies. The term “rapid” is used because reduction in the volume of data acquisition leads to faster scans. The objective is to obtain high acceleration factors, since it indicates the ability of the technique to yield high-quality images with reduced data (in turn, reduced acquisition time). Reduced data acquisition in conventional (sequential) MR scanners, where a single receiver coil is used, can be achieved either by acquiring only certain k-space regions or by regularly undersampling the entire data in k-space. In parallel MR scanners, where multiple receiver coils are used to acquire high-SNR data, reduced data acquisition is typically accomplished using regular undersampling. Optimal region selection in the 3D k-space (restricted to ky - kz plane, since kx is the readout direction) needs to satisfy “maximum energy compaction” and “minimum acquisition” requirements. In this thesis, a novel star-shaped truncation window is proposed to increase the achievable acceleration factor. The proposed window monotonically cuts down the acquisition of the number of k-space samples with lesser energy. The truncation window samples data within a star-shaped region centered around the origin in the ky - kz plane. The missing values are extrapolated using generalized series modeling-based methods. The proposed method is applied to several real and synthetic data sets. The superior performance of the proposed method is illustrated using the standard measures of error images and uptake curve comparisons. Average values of slope error in estimating the enhancement curve are obtained over 5 real data sets of breast and abdomen images, for an acceleration factor of 8. The proposed method results in a slope error of 5%, while the values obtained using rectangular and elliptical windows are 12% and 10%, respectively. k-t BLAST, a popular method used in cardiac and functional brain imaging, involves regular undersampling. However, the method suffers from drawbacks such as separate training scan, blurred training estimates and aliased phase maps. In this thesis, variations to k-t BLAST have been proposed to overcome the drawbacks. The proposed improved k-t BLAST incorporates variable-density sampling scheme, phase information from the training map and utilization of generalized-series extrapolated training map. The advantage of using a variable density sampling scheme is that the training map is obtained from the actual acquisition instead of a separate pilot scan. Besides, phase information from the training map is used, in place of phase from the aliased map; generalized series extrapolated training map is used instead of the zero-padded training map, leading to better estimation of the unacquired values. The existing technique and the proposed variations are applied on real fMRI data volumes. Improvement in PSNR of activation maps of up to 10 dB. Besides, a reduction of 10% in RMSE is obtained over the entire time series of fMRI images. The peak improvement of the proposed method over k-t BLAST is 35%, averaged over 5 data sets. Most image reconstruction techniques in parallel MR imaging utilize the knowledge of coil sensitivities for image reconstruction, along with assumptions of image reconstruction functions. The thesis proposes an image reconstruction technique that neither needs to estimate coil sensitivities nor makes any assumptions on the image reconstruction function. The proposed cartesian parallel imaging using neural networks, called “Composite image Reconstruction And Unaliasing using Neural Networks” (CRAUNN), is a novel approach based on the observation that the aliasing patterns remain the same irrespective of whether the k-space acquisition consists of only low frequencies or the entire range of k-space frequencies. In the proposed approach, image reconstruction is obtained using the neural network framework. Data acquisition follows a variable-density sampling scheme, where low k-space frequencies are densely sampled, while the rest of the k-space is sparsely sampled. The blurred, unaliased images obtained using the densely sampled low k-space data are used to train the neural network. Image is reconstructed by feeding to the trained network, the aliased images, obtained using the regularly undersampled k-space containing the entire range of k-space frequencies. The proposed approach has been applied to the Shepp-Logan phantom as well as real brain MRI data sets. A visual error measure for estimating the image quality used in compression literature, called SSIM (Structural SIMilarity) index is employed. The average SSIM for the noisy Shepp-Logan phantom (SNR = 10 dB) using the proposed method is 0.68, while those obtained using GRAPPA and SENSE are 0.6 and 0.42, respectively. For the case of the phantom superimposed with fine grid-like structure, the average SSIM index obtained with the proposed method is 0.7, while those for GRAPPA and SENSE are 0.5 and 0.37, respectively. Image reconstruction is more challenging with reduced data acquired using non-cartesian trajectories since aliasing introduced is not localized. Popular technique for non-cartesian parallel imaging CGSENSE suffers from drawbacks like sensitivity to noise and requirement of good coil estimates, while radial/spiral GRAPPA requires complete identical scans to obtain reconstruction kernels for specific trajectories. In our work, the proposed neural network based reconstruction method, CRAUNN, has been shown to work for general non-cartesian acquisitions such as spiral and radial too. In addition, the proposed method does not require coil estimates, or trajectory-specific customized reconstruction kernels. Experiments are performed using radial and spiral trajectories on real and synthetic data, and compared with CGSENSE. Comparison of error images shows that the proposed method has far lesser residual aliasing compared to CGSENSE. The average SSIM index for reconstructions using CRAUNN with spirally and radially undersampled data, are comparable at 0.83 and 0.87, respectively. The same measure for reconstructions using CGSENSE are 0.67 and 0.69, respectively. The average RMSE for reconstructions using CRAUNN with spirally and radially undersampled data, are comparable at 11.1 and 6.1, respectively. The same measure for reconstructions using CGSENSE are 16 and 9.18, respectively.