In this paper, we provide a review of polarimetric synthetic aperture radar applications and analytical algorithms that are necessary tools for these applications. We first introduce coherent polarimetric optimization algorithms, then some for the partially coherent case, compare three frequently applied target matrix decomposition algorithms, and show their major differences. Topics of polarization orientation angle estimation and speckle filtering are also discussed and illustrated with examples. The important applications of terrain and land-use classification are described from the concept of the maximum likelihood classifier based on the Wishart distribution. In addition, recently developed techniques focused on soil moisture, surface roughness, polarimetric SAR interferometry and SAR tomography are outlined. Most of the topics discussed are interrelated, and these comparative assessments are discussed throughout this paper