The spreading of Newtonian fluids on smooth solid substrates is well understood; the speed of the contact line is given by a competition between capillary driving forces and viscous dissipation, yielding Tanner's law $R \propto t^{1/10}$. Here we study the spreading of non-Newtonian liquids, focusing on the two most common non-Newtonian flow properties, a shear-rate dependence of the viscosity and the existence of normal stresses. For the former, the spreading behaviour is found not to deviate strongly from Tanner's law. This is quite surprising given that, within the lubrication approximation, it can be shown that the contact line singularity disappears due to the shear-dependent viscosity. The experiments are compared with the predictions of the lubrication theory of power-law fluids. If normal stresses are present, again only small deviations from Tanner's law are found in the experiment. This can be understood by comparing viscous and normal stress contributions to the spreading; it turns out that only logarithmic corrections to Tanner's law survive, which are nonetheless visible in the experiment.