There have been a few videos around since as far back as 2010 that people can run on water. This is less commonly called Liquid Mountaineering. This is apparently a new sport, but I have only ever seen videos of people doing this, e.g Walk on water (Liquid Mountaineering).
The Mythbusters tried this and they came to the conclusion that this myth was "busted", but as we should all know, we should not take what these guys do as a bible. The myth here: Episode 162: Running on Water.
There seems to be a lot of debate over this in comments of the running on water videos, as you can hide a platform under the water which will make you look like you are running on water, for example. This means it could be a simple illusion of running on water rather than actually running on water.
Definition of "Running on water": Running on water is the art of running un-supported on water, to remain on top of the water for a given series of steps on the water.