In many applications optical wavefields with reduced states of coherence are preferable to fully coherent fields. Gaussian Schell-model beams can be viewed as generalizations of the ordinary fundamental Gaussian laser beam by allowing a Gaussian variation of the transverse spatial coherence. Suitable propagation parameters for Gaussian Schell-model beams are stated, and their evolution in free space and through optical systems is analyzed. An illustrative geometrical approach, analogous to the propagation-circle method for laser beams, is also presented. Finally, the effects of spatial coherence on beam focusing are investigated, and applications of the algebraic and graphical methods to some particular problems are briefly discussed.