In this paper, over fixed directed communication graph, the leader-following consensus of heterogenous fractional-order multi-agent systems (HFOMASs) is investigated with respect to input delays, where the fractional orders between leader and followers are heterogenous, which is more general. Firstly, a control algorithm with a fractional-order estimator is proposed to guarantee the leader-following consensus of the HFOMASs. Then, the identical input delays are taken into account in the above control algorithm, and the leader-following consensus of the HFOMASs can be achieved under the derived sufficient and necessary condition. Thirdly, the diverse input delays are further considered in the HFOMASs, and a sufficient condition is put forward under the designed control algorithm. Finally, simulations are conducted to make the results be convinced.