New techniques for remote sensing of solar magnetic fields now provide measures of the magnetic field vector within the solar atmosphere with high angular resolution and high precision. These measurements have enabled a much improved physical understanding of magnetic processes and phenomena in the solar atmosphere, processes that drive the variability of the Sun's radiative and particulate output. The new techniques are reviewed here in the context of the scientific advances they have fostered. Emphasis is given to techniques for inferring the field vector. The quantitative nature of the information needed to explore the solar phenomena sharply constrains the needed precision and angular resolution of the observations. These requirements are reviewed here, along with an assessment of how future improvements in observing capabilities will address these requirements. One may also attribute much of the recent advance in our understanding of solar magnetic fields to ongoing progress in techniques for analysis of the polarization measurements that underlie solar magnetometry. The status and prospects of analysis techniques are also reviewed. ¿ 2000 American Geophysical Union