Sprinting is a feature introduced in GTA III and is continued in all games after that.
Sprinting allows the player to run faster. In order to sprint on Playstation, X must be held down. Xbox requires A to be held and on PC, Left or Right Shift is usually used. Sprinting is not unlimited however, after a certain amount of time the player will tire and will not be able to sprint for a set amount of time. If standing still, the player will show their exhaustion. The sprint ability was carried on in GTA IV, Lost and Damned and The Ballad of Gay Tony. In the GTA III Era, if the Paramedic side mission is completed, unlimited sprint is unlocked, excluding in GTA San Andreas which it can only be unlocked by completing Burglar side mission with $10,000.
In GTA IV you can sprint (instead of jog) by tapping the sprinting button quickly instead of holding it down.