Four methods for designing digital filters optimal in the Chebyshev sense are developed. The properties of these filters are investigated and compared. An analytic method for designing narrow-band FIR filters using Zolotarev polynomials, which are extensions of Chebyshev polynomials, is proposed. Bandpass and bandstop narrow-band filters as well as lowpass and highpass filters can be designed by this method. The design procedure, related formulae and examples are presented. An improved method of designing optimal minimum phase FIR filters by directly finding zeros is proposed. The zeros off the unit circle are found by an efficient special purpose root-finding algorithm without deflation. The proposed algorithm utilizes the passband minimum ripple frequencies to establish the initial points, and employs a modified Newton's iteration to find the accurate initial points for a standard Newton's iteration. The proposed algorithm can be used to design very long filters (L = 325) with very high stopband attenuations. The design of FIR digital filters in the complex domain is investigated. The complex approximation problem is converted into a near equivalent real approximation problem. A standard linear programming algorithm is used to solve the real approximation problem. Additional constraints are introduced which allow weighting of the phase and/or group delay of the approximation. Digital filters are designed which have nearly constant group delay in the passbands. The desired constant group delay which gives the minimum Chebyshev error is found to be smaller than that of a linear phase filter of the same length. These filters, in addition to having a smaller, approximately constant group delay, have better magnitude characteristics than exactly linear phase filters with the same length. The filters have nearly equiripple magnitude and group delay. The problem of IIR digital filter design in the complex domain is formulated such that the existence of best approximation is guaranteed. An efficient and numerically stable algorithm for the design is proposed. The methods to establish a good initial point are investigated. Digital filters are designed which have nearly constant group delay in the passbands. The magnitudes of the filter poles near the passband edge are larger than of those far from the passband edge. A delay overshooting may occur in the transition band (don't care region), and it can be reduced by decreasing the maximum allowed pole magnitude of the design problem at the expense of increasing the approximation error.