This dissertation develops new techniques employing the Low-rank and Sparse Representation approaches to improve the performance of state-of-the-art algorithms in hyperspectral image analysis. The contributions of this dissertation are outlined as follows. 1) Low-rank and sparse representation approaches, i.e., low-rank representation (LRR) and low-rank subspace representation (LRSR), are proposed for hyperspectral image analysis, including target and anomaly detection, estimation of the number of signal subspaces, supervised and unsupervised classification. 2) In supervised target and unsupervised anomaly detection, the performance can be improved by using the LRR sparse matrix. To further increase detection accuracy, data is partitioned into several highly-correlated groups. Target detection is performed in each group, and the final result is generated from the fusion of the output of each detector. 3) In the estimation of the number of signal subspaces, the LRSR low-rank matrix is used in conjunction with direct rank calculation and soft-thresholding. Compared to the state-of-the-art algorithms, the LRSR approach delivers the most accurate and consistent results across different datasets. 4) In supervised and unsupervised classification, the use of LRR and LRSR low-rank matrices can improve classification accuracy where the improvement of the latter is more significant. The investigation on state-of-the-art classifiers demonstrate that, as a pre-preprocessing step, the LRR and LRSR produce low-rank matrices with fewer outliers or trivial spectral variations, thereby enhancing class separability.