This paper considers the design of a digital filter with prescribed magnitude and group delay specifications. First, we outline the derivation of the phase and group delay functions of an Nth order digital Laguerre filter and show that the group delay function of the filter can be written as a ratio of quadratic functions in the filter coefficients. Then, we formulate our filter design problem as a constrained L2 space minimization problem in which the performance requirement on the group delay and magnitude in the passband are treated as constraints while minimizing the L2 norm of the error function between the designed and the desired filters. Methods for solving the proposed nonlinear optimization problem axe outlined. Numerical results are presented to illustrate the usefulness of the proposed method. As a special case, corresponding results for general FIR filters axe also derived