This paper deals with the leader-following consensus problem for a class of multi-agent systems with nonlinear dynamics and directed communication topology. Two distributed adaptive nonlinear control laws are proposed based on the relative state information between neighboring agents, which solve the leader-following consensus problem and the inverse optimal adaptive leader-following consensus problem, respectively. Compared with the existing results in the liter-ature, we do not put any restrictions on nonlinear functions and the adaptive consensus protocols are in a fully distributed fashion. A numerical example is given to verify the theoretical analysis.