This thesis presents a new approach for source localization using sensor arrays based on sparse signal reconstruction. The core idea is to represent the source localization problem as an inverse problem and apply sparsity-enforcing regularization to achieve high resolution and robustness to noise and limited data. The approach involves reformulating the problem in an optimization framework using an overcomplete basis and applying sparsity regularization to focus signal energy. Numerical methods for enforcing sparsity are developed using $ \ell_{1} $ and $ \ell_{p} $ regularization. The $ \ell_{1} $ regularization is solved using second-order cone programming, while the $ \ell_{p} $ regularization is addressed using half-quadratic regularization. The approach is extended to allow self-calibration of sensor position errors by incorporating both source and sensor positions into an augmented objective function. The method is also shown to have important advantages such as superresolution, robustness to noise and limited data, and robustness to correlated sources. The thesis also includes a theoretical analysis of the noiseless signal representation problem using overcomplete bases, showing the uniqueness of solutions to the $ \ell_{0} $ problem and the equivalence of $ \ell_{0} $, $ \ell_{1} $, and $ \ell_{p} $ problems under certain sparsity conditions. The work is supported by extensive numerical experiments comparing the proposed method to existing source localization techniques, demonstrating its effectiveness in various scenarios. The thesis also addresses practical issues such as the choice of regularization parameters, the effects of the grid, and the number of resolvable sources. The results show that the sparse regularization framework provides significant improvements in resolution and robustness compared to traditional methods. The work has important implications for applications in wireless communications, radar, sonar, and exploration seismology.This thesis presents a new approach for source localization using sensor arrays based on sparse signal reconstruction. The core idea is to represent the source localization problem as an inverse problem and apply sparsity-enforcing regularization to achieve high resolution and robustness to noise and limited data. The approach involves reformulating the problem in an optimization framework using an overcomplete basis and applying sparsity regularization to focus signal energy. Numerical methods for enforcing sparsity are developed using $ \ell_{1} $ and $ \ell_{p} $ regularization. The $ \ell_{1} $ regularization is solved using second-order cone programming, while the $ \ell_{p} $ regularization is addressed using half-quadratic regularization. The approach is extended to allow self-calibration of sensor position errors by incorporating both source and sensor positions into an augmented objective function. The method is also shown to have important advantages such as superresolution, robustness to noise and limited data, and robustness to correlated sources. The thesis also includes a theoretical analysis of the noiseless signal representation problem using overcomplete bases, showing the uniqueness of solutions to the $ \ell_{0} $ problem and the equivalence of $ \ell_{0} $, $ \ell_{1} $, and $ \ell_{p} $ problems under certain sparsity conditions. The work is supported by extensive numerical experiments comparing the proposed method to existing source localization techniques, demonstrating its effectiveness in various scenarios. The thesis also addresses practical issues such as the choice of regularization parameters, the effects of the grid, and the number of resolvable sources. The results show that the sparse regularization framework provides significant improvements in resolution and robustness compared to traditional methods. The work has important implications for applications in wireless communications, radar, sonar, and exploration seismology.