Adaptive filter theory is an important subject in the field of signal processing and has numerous applications in speech processing, telecommunications and mechanical engineering. Examples in speech processing include suppression of background noise, echo cancellation and dereverberation (acoustic path equalization). Examples in telecommunications include channel equalization and interference cancellation. Subband filter banks are introduced in the area of adaptive filtering in order to improve the performance of time domain adaptive filters. The main performance improvements are faster convergence speed and reduction of computational complexity due to shorter adaptive filters in the subbands. The thesis deals with design and analysis of subband adaptive systems using subband filter banks, with applications in speech enhancement. Multirate subband filter banks often introduce signal degradations. Some of these degradations are inherent from the multirate building blocks and some are due to improperly designed analysis and synthesis filters. Filter banks need to be designed in such a way, that the benefits are sustained while degradations due to the multirate processing are minimized. The presented design methods in this thesis aim to address two important filter bank properties, transmission delay and aliasing distortions caused by multi-rate building blocks. Aliasing distortions appear in the subband signals and in the reconstructed output signal, and have great impact on the performance of the speech enhancement algorithms as well as on the quality of the output enhanced speech. The main contributions in this thesis are design methods for uniform and nonuniform filter banks for adaptive filtering applications. The proposed methods are compared with conventional methods, and performance aspects of different filter bank designs are compared in different speech processing applications. These applications include acoustic echo cancellation, speech enhancement, and subband beamforming. Real speech signals are used in evaluations and results show that properly designed filter banks play an important role in the quality of the output enhanced speech. Another important contribution is the convergence analysis of certain types of delayless subband adaptive filters. The analysis leads to design requirements for improved convergence speed of such adaptive filters.