Compressed sensing (CS) is an emerging signal acquisition framework that goes against the traditional Nyquist sampling paradigm. CS demonstrates that a sparse, or compressible, signal can be acquired using a low rate acquisition process. Since noise is always present in practical data acquisition systems, sensing and reconstruction methods are developed assuming a Gaussian (light-tailed) model for the corrupting noise. However, when the underlying signal and/or the measurements are corrupted by impulsive noise, commonly employed linear sampling operators, coupled with Gaussian-derived reconstruction algorithms, fail to recover a close approximation of the signal. This dissertation develops robust sampling and reconstruction methods for sparse signals in the presence of impulsive noise. To achieve this objective, we make use of robust statistics theory to develop appropriate methods addressing the problem of impulsive noise in CS systems. We develop a generalized Cauchy distribution (GCD) based theoretical approach that allows challenging problems to be formulated in a robust fashion. Robust sampling operators, together with robust reconstruction strategies are developed using the introduced GCD framework. To solve the problem of impulsive noise embedded in the underlying signal prior the measurement process, we propose a robust nonlinear measurement operator based on the weighed myriad estimator. To recover sparse signals from impulsive noise introduced in the measurement process, a geometric optimization problem based on L1 minimization employing a Lorentzian norm constraint on the residual error is introduced. Additionally, robust reconstruction strategies that incorporate prior signal information into the recovery process are developed. First, we formulate the sparse recovery problem in a Bayesian framework using probabilistic priors from the GCD family to model the signal coefficients and measurement noise. An iterative reconstruction algorithm is developed from this Bayesian framework. Second, we develop a Lorentzian norm based iterative hard thresholding algorithm capable of incorporating prior support knowledge into the recovery process. The derived algorithm is a fast method capable of handling large scale problems whilst having robustness against impulsive noise. Analysis of the proposed methods shows that in impulsive environments, when the noise posses infinite variance, a finite reconstruction error is achieved and furthermore these methods yield successful reconstruction of the desired signal. Experimental results demonstrate that the proposed methods significantly outperform commonly employed compressed sensing sampling and reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments. Simulation results also show that the proposed algorithms with prior signal information require fewer samples than most existing reconstruction methods to yield approximate reconstruction of sparse signals. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]