The NEXRAD radars have provided information on storm structure and other atmospheric phenomena never before available to operational forecasters. The radars can scan the atmosphere at high spatial and temporal resolutions, yet the measurements are limited mainly to reflectivity and radial-velocity component (along the radar beam) and the data are not free of quality problems. Using radar observations to improve numerical weather predictions requires major research efforts in radar data quality control and assimilation. Applied mathematical problems encountered in the required research are often critical but not completely solved. Three examples will be presented in this talk. The first example involves a global minimization problem in which multiple local minima are caused by the zigzag-discontinuities of modified observation operator in the cost-function. The modified observation operator mimics the aliasing effect in raw radial-velocity data. The second example presents the methods developed for wind error covariance estimation and vector wind estimation in our current real-time radar wind analysis system. The third example shows the singular-value and spectral formulations derived for measuring information content from observations and their applications to radar data compression and radar scan optimization.