The men’s biathlon race was originally scheduled for Sunday evening but organisers had pushed it back to Monday morning.
A decision on whether the women's race, scheduled on Monday at 1900 local time (1500GMT/11 AM ET), will take place, is to be made at 1600 local time.
Biathlon combines cross-country skiing and rifle shooting. Athletes have to complete a distance over skis and stop at a shooting range two or four times depending on the discipline.
As thick fog covered the mountains on Monday a view of the targets - 50 metres from where athletes stand or lie - was impossible at the Laura centre.
It was the third time that a biathlon race at the Olympics has been postponed.
At the Nagano Games in 1998 the men's sprint was called off after heavy snow and fog forced the race to be stopped. It was started again the following day.
In Sapporo in 1972, the same scenario applied to the men's 20km individual.
France's Martin Fourcade, who has already claimed two gold medals in Sochi, and Norway's Ole Einar Bjoerndalen, who is looking to secure a record-breaking 13th Winter Olympics medal, are among the favourites for the men's mass start.
The men's snowboard cross competition will now be run on Tuesday after thick fog in the Caucasus mountains forced Olympic organisers to postpone it on Monday.
The event will take place with a shortened format at the Rosa Khutor Extreme Park at 1030 local time (0630 GMT) on Tuesday.
There will be no seeding runs and the competition will go straight to the first knockout round with riders seeded according to their world rankings.
Fog and drizzle shrouded the course on Monday morning, reducing visibility and making it highly unlikely the event would get underway as scheduled.
After several delays, conditions had not improved sufficiently and organisers decided to call off the event for the day just after 2pm local time.
"We were inspecting the course today in the morning, but I couldn't see more than 10 metres ahead," German rider Konstantin Schad told reporters.
"I guess the maximum speed we reach is 80 kmh. There is no way to compete in fog, it's much too dangerous," he said.
Safety concerns were to the fore after a slew of serious injuries at the Rosa Khutor Extreme Park over the weekend.
In the worst of these, Russian ski cross athlete Maria Komissarova underwent back surgery following a serious crash in training on Saturday.