The field-effect mobility in thin-film transistors based on alpha-sexithiophene (alpha-6T) and related materials displays a temperature dependence that is remarkably nonmonotonic. Above a transition temperature T-T (specific to a given material) the transport is thermally activated, whereas below T-T there is a very steep enhancement of the mobility. In the activated regime, the results are well described by the theoretical predictions for small polaron motion made by Holstein in 1959. An analysis of the transistor characteristics shows that the hopping transport in these devices is intrinsic. Performance limits for devices based on alpha-6T and related materials were established; these limits point to the strong possibility that better molecular materials for transistor applications may be designed from first principles.