We study the transition zone or contact line between a thin film and bulk liquid, and calculate the line tension tau, employing an interface displacement model equivalent to Derjaguin's and de Gennes' approach. We investigate the behaviour Of tau in the limit that the contact angle theta tends to zero, approaching a wetting phase transition. Previous results for wetting and prewetting, derived in the gradient-squared approximation of the model, remain valid when the gradient is included to all orders. The interesting singular behaviour Of tau at wetting is universal, due to the critical phenomena that have recently been found to underly first-order as well as continuous wetting transitions. Finally we critically review two contact-line instabilities that have been associated with contact-angle hysteresis, and propose an alternative physical interpretation.