We present here a new method for realization ofminimal-order proper rational transfer functions from z-domain samples. A criterion for determining the minimal order of the transfer function is given. The theory is developed first for strictly proper rational transfer functions and then extended to the case of transfer functions with a direct-coupling term. The theory is valid for arbitrary z-domain samples. The special case of uniform frequency response samples is shown to allow a computationally efficient derivation of the realization. Examples are given to illustrate the theory. The theory developed here has many applications, such as modelling of communication channels, identification of systems from frequency measurements and infinite impulse response filter design.