This paper presents optimum and sub-optimal designs of noise-shaping FIR filters for single- and multi-bit data converters. In the designs, only three parameters, the number of taps, oversampling ratio (OSR) and l1-norm of the filter coefficients are specified, and the in-band peak of the amplitude response is minimized under the specifications. The minimization problem is formulated with the overload-free condition, which guarantees the rigorous stability, and an overload-free converter generates no distortion in any output signals. In the optimum design, the minimization problem is directly and exactly solved, but the sub-optimal method solves this problem by iteratively utilizing the simplex method. The iterative sub-optimal method without the exact optimality is far faster and more efficient than the optimum method. In design examples, optimum and sub-optimal noise-shaping FIR filters for single- and multi-bit data converters are designed, and their optimal performance is revealed. For single-bit data converters with OSR 64, a noise-shaping FIR filter is designed and then shown to achieve a signal to noise and distortion ratio (SNDR) 107.6 [dB] in the band of interest.