Travelling around for quite a while and sometimes, well, just following the sun, today the question occurred to me: What happens if you really do this?
So let's say some point is moving along the earth's surface, with constant velocity, at each time heading in that direction in which the sun appears to it (as long as the sun is visible to it; let's say it is motionless from local sunset to local sunrise, and also when locally the sun is in zenith). I am fine with the idealisation that the earth is a sphere, but if possible I would like to have the earth's orbit as an ellipse, and the axis tilt in consideration. (Relativistic effects on the other hand as well as axial precession, nutation and the like do not have to be considered.) The answer obviously would have as parameters the chosen velocity and (the latitude of) the starting point. Graphics would probably be nice.
First of all, if you are going in the direction of the sun, you won't remain on the same latitude (due the tilt of the earth). You keep on going down (if you are near the North Pole) or up (if you are near the South Pole) till you reach a latitude where you would go down and then up equally; thus reaching back to the starting latitude. This latitude could be anywhere on the earth and would vary upon the date of the year. As the date changes, so will the latitude where you can possibly return to.
As for your speed, the perfect speed required would vary depending on your place on the earth. If you travel at a constant velocity, you will probably either not be able to catch up with the sun, or find it overhead and wait (or reach the next night, in rare circumstances, wherein you'll have to wait till the sun rises for the place on the earth).
I'm not 100% sure though.