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This paper discusses the use of Capon's minimum-variance method (MVM) and Amplitude and Phase EStimation (APES) spectral-estimation algorithms to synthetic aperture radar range-azimuth focusing. The rationale of the algorithms is discussed. An implementation of a Capon or APES processing chain is explained, and processing parameters such as chip-image size, resampling factor, and diagonal loading are discussed. For multichannel cases, a joint-processing approach is presented. A set of Monte Carlo simulations are described and used to benchmark Capon- and APES-based processing against conventional matched-filter-based approaches. Both methods improve the resolution and reduce sidelobes. APES yields generally better estimates of amplitude and phase than Capon but with worse resolution. Results with RADARSAT-2 quad-polarization data over Barcelona are used to qualitatively study the real-life performance of these algorithms.