Adaptive filters are considered nonlinear systems; therefore their behavior analysis is more complicated than for fixed filters. As adaptive filters are self designing filters, their design can be considered less involved than in the case of digital filters with fixed coefficients. The paper discusses adaptive filters, adaptive filtering with various approaches, optimization methods, algorithms for a filter, IIR and FIR filter designs, in order to improve a prescribed performance criterion. Further least Pth norm approach is compared for both FIR and IIR filter. Since no specifications are available, the adaptive algorithm that determines the updating of the filter coefficients requires extra information that is usually given in the form of a signal. This signal is in general called a desired or reference signal, whose choice depends on the application.