Using a validated general circulation model, we determine where and for how long the surface pressure and surface temperature on Mars meet the minimum requirements for the existence of liquid water in the present climate system: pressures and temperatures above the triple point of water but below the boiling point. We find that for pure liquid water, there are five ``favorable'' regions where these requirements are satisfied: between 0° and 30°N in the plains of Amazonis, Arabia, and Elysium; and in the Southern Hemisphere impact basins of Hellas and Argyre. The combined area of these regions represents 29% of the planet's surface area. In the Amazonis region these requirements are satisfied for a total integrated time of 37 sols each Martian year. In the Hellas basin the number of degree days above zero is 70, which is well above those experienced in the dry valley lake region of Antarctica. These regions are remarkably well correlated with the location of Amazonian paleolakes mapped by Cabrol and Grin  but are poorly correlated with the seepage gullies found by Malin and Edgett . In both instances, obliquity variations may play a role. For brine solutions the favorable regions expand and could potentially include most of the planet for highly concentrated solutions. Whether liquid water ever forms in these regions depends on the availability of ice and heat and on the evaporation rate. The latter is poorly understood for low-pressure CO2 environments but is likely to be so high that melting occurs rarely, if at all. However, even rare events of liquid water formation would be significant since they would dominate the chemistry of the soil and would have biological implications as well. It is therefore worth reassessing the potential for liquid water formation on present day Mars, particularly in light of recent Mars Global Surveyor observations.