A single kinesin molecule can move "processively" along a microtubule for more than 1 micrometer before detaching from it. The prevailing explanation for this processive movement is the "walking model," which envisions that each of two motor domains (heads) of the kinesin molecule binds coordinately to the microtubule. This implies that each kinesin molecule must have two heads to "walk" and that a single-headed kinesin could not move processively. Here, a motor-domain construct of KIF1A, a single-headed kinesin superfamily protein, was shown to move processively along the microtubule for more than 1 micrometer. The movement along the microtubules was stochastic and fitted a biased Brownian-movement model.