The theory of stellar winds as given by the equations of classical fluid dynamics is considered. The equations of momentum and energy describing a steady, spherically symmetric, heat-conducting, viscous stellar wind are cast in a dimensionless form which involves a thermal conduction parameter E and a viscosity parameter &ggr;. An asymptotic analysis is carried out, for fixed &ggr;, in the cases E→O and E→∞ (corresponding to small and large thermal conductivity, respectively), and it is found that it is possible to construct critical solutions for the wind velocity and temperature over the entire flow. The E→O solution represents a wind which emanates from the star at low, subsonic speeds, accelerates through a sonic point, and then approaches a constant asymptotic speed, with its temperature varying as r-4/3 at large distances r from the star; the E→∞ solution represents a wind which, after reaching an approximately constant speed, with temperature varying as r-2/7, decelerates through a diffuse shock and approaches a finite pressure at infinity. A categorization is made of all critical stellar wind solutions for given values of &ggr; and E, and actual numerical examples are given. Numerical solutions are obtained by integrating upstream 'from infinity' from initial values of the flow parameters given by appropriate asymptotic expansions. The role of viscosity in stellar wind theory is discussed, viscous and inviscid stellar wind solutions are compared, and it is suggested that with certain limitations, the theory presented may be useful in analyzing winds from solar-type stars.