And I am not really looking into some summer blockbusters (2012/Day after tomorrow and other Emmerich movies) where one hero saves a day (but please no spoilers if something has good or bad ending) but something more intelligent, less action movie.

What quickly comes to my mind:

Blindness

Contagion / Outbreak

Threads / The Day After

Mentioned by other users:

On the Beach 1959

Last Night 1998

The Happening (2008)

The Trigger Effect (1996)

The Road

Seeking a Friend for the End of the World

Children of Men 2006

The Survivalist 2015

Childhoods End

Twelve monkeys

Melancholia