Can a person or animal run on the surface of water?

Partly settled. Some of it is understood, some is still argued about.

Kapitza's problem #18 asks for the minimum speed needed to run across a water surface without sinking. The mechanism in basilisk lizards (slap-stroke-retraction) was quantified by Glasheen & McMahon (1996, Nature) and later resolved into full 3D force measurements by Hsieh & Lauder (2004, PNAS). Applying the same impulse-momentum model to human-scale bodies shows unaided human water-running is infeasible at Earth gravity but becomes possible at Moon-like gravity or with rigid fin extensions, confirmed experimentally in a reduced-gravity pool rig in 2012.

A tiny lizard exploits pure Newtonian impact mechanics to defeat gravity for a few tenths of a second per stride, and the same physics tells you exactly why a person cannot do it, and exactly what would have to change for them to.

Explanations that circulated

• Basilisks 'skate' via surface tension alone (surface tension is far too weak at lizard/human size; the effect is impact-driven hydrodynamic force from slapping, not capillarity)
• Humans could run on water with strong enough legs at 1g (models show it requires either much lower gravity or artificial fins, not raw strength)

References

• Running on water: Three-dimensional force generation by basilisk lizards, S. Tonia Hsieh, George V. Lauder, PNAS, 2004
• Humans Running in Place on Water at Simulated Reduced Gravity, Various (adapted Glasheen & McMahon model), PLOS ONE, 2012