In many signal processing and communication (SPC) applications we require to estimate signal corrupted by channel and additive noise. Optimal linear filters and predictors are used to recover signal from given observed (corrupted) signal. Kalman and Wiener filters are commonly used as optimal filters. These filters minimize the mean square error (MSE) or variance of the output error. The minimization require exact knowledge of input signal and noise power spectral density (PSD). Therefore, the performance of Kalman or Wiener filters degrade if the input signal and noise statistics is changing with time and is not known a priori. In many SPC applications there is no exact knowledge of the input signal and noise Statistics and Probability; One solution to this is to use the filters which minimizes MSE and adapt to changing input signals and noise Statistics and Probability; This solution falls into a general category of adaptive filters. Often, convergence speed of the adaptive filter algorithm determines the performance as it is assumed that the convergence speed is fast enough to track the changes in the input signal and noise Statistics and Probability; If the convergence speed is not able to track the input signal and noise statistics one can expect large variation in the output error power. Another approach to overcome unknown input signal and noise statistics is to use the mini-max estimation. One approach towards mini-max estimation is to minimize the error using H[infinity] criteria to obtain H[infinity] filters. This will lead to a conservative (minimize over the worst case input signals) design that is more robust to changes in the input signal and noise Statistics and Probability;;In this dissertation, interpretation of H[infinity] filters for zero mean stationary signals is discussed. From this H[infinity] filters are represented in the time and frequency domain. Performance benefits of H[infinity] filters over minimum variance filters are derived from this representation. Mathematical solutions to compute sub-optimal H[infinity] filters in time and frequency domain are discussed. Finally, performance benefits of H[infinity] filters for the code division multiple access (CDMA) system, signal estimation problems, and adaptive filters are shown through simulation results.