By no means a comprehensive list, just 13 of my favorite stories dealing with the end-of-the-world-as-we-know-it. Thirteen seemed an appropriate number given the subject matter.
"By the Rivers of Babylon" by Stephen Vincent Benet (short story) Written in 1937, this is one of the first apocalyptic pieces of fiction I remember reading in 9th grade. It blew me away. A character named John narrates the tale of his exploration of the forbidden "Place of the Gods" in a world that has nearly forgotten the existence of 20th century civilization. John is the son of a priest of a tribe of hunters, heirs to a global catastrophe, whose curiosity takes him on a journey of discovery and search for truth about his civilized ancestors and the statue of a god called ashington.
A Canticle for Leibowitz by Walter M. Miller, Jr. (novel) Set in a Roman Catholic monastery in the desert of the Southwestern United States after a devastating nuclear war, the story spans thousands of years as civilization rebuilds itself. The monks of the Albertian Order of Leibowitz take up the mission of preserving the surviving remnants of man's scientific knowledge until the day the outside world is again ready for it.
Alas, Babylon by Pat Frank (novel) It was one of the first post-apocalyptic novels of the nuclear age and remains popular fifty years after it was first published. The novel deals with the effects of a nuclear war on the small town of Fort Repose, Florida. David Brin has written that his novel was highly influential for him as he wrote The Postman (see later down the list.)
Emergence by David R. Palmer (novel) Candy is a young girl with a high I.Q., a natural talent for martial arts and the ability to perform quick bursts of near superhuman activity. One day, her father is called to Washington to deal with a mysterious problem. While he is gone, a disease wipes out most of humanity. Candy lives at her house for a while, as she has many emergency supplies. She soon realizes that she must venture outside. She decides to go exploring for supplies and to figure out what happened. Her original companion is a loyal parrot but later she meets other survivors.
Dhalgren by Samuel R. Delany (novel) A difficult book to read due to a lack of lineal plot or consistent narrative, reminiscent of Thomas Pynchon's Gravity's Rainbow. Even now I can't really say I enjoyed the book, but for its sheer scope and virtuosity, I have to put it on the list.
The Sheep Look Up by John Brunner (novel) With the rise of a corporation-sponsored government (hmmm), pollution in big cities has reached extreme levels and most (if not all) people's health has been affected in some way. Similar to Atlas Shrugged due to its uncanny prediction of our current state of affairs around the world.
Cat's Cradle by Kurt Vonnegot (novel). A dead-on satire of the world turned to water by Ice-Nine. Vonnegut gets his well-aimed shots in at science, technology, the arms race and religion.
I Am Legend by Richard Matheson (novel) The story of Robert Neville who is apparently the sole survivor of a bacterial pandemic apocalypse the symptoms of which resemble vampirism. The book details Neville's daily life in Los Angeles, as he attempts to comprehend, research, and possibly cure the disease that killed mankind, and to which he is immune (Neville assumes this is because he was bitten by a vampire bat who was "infected". Because it was not a human, it did not kill Neville, instead, he became ill for a period of time). Much of the book deals with Neville's emotional struggle to cope with losing his humanity is dealt with by going about a daily routine. Turned into 3 successful movies. The Last Man Left with Vincent Price; The Omega Man with Charlton Heston and the recent Will Smith sci-fi thriller. Great, great novel.
Lucifer's Hammer by Larry Niven and Jerry Pournelle (novel) And long and epic story that details a cometary impact on Earth, the end of civilization, and the battle for the future. It encompasses the discovery of the comet, the LA social scene, and a cast of diverse characters whom fate seems to smile upon and allow to survive the massive cataclysm and the resulting tsunamis, plagues, famines and battles amongst scavengers and cannibals.
The Postman by David Brin (novel) Years after a cataclysmic event, a drifter, Gordon Crantz, stumbles across the uniform of an old United States Postal Uniform letter carrier and gives hope to a community threatened by local hypersurvivalists warlords with empty promises of aid from the "Restored United States of America". The basis of the REALLY BAD Kevin Costner movie of the same name. DO NOT LET THAT KEEP YOU FROM READING THE BOOK.
Swan Song by Robert McCammon (novel) Winner of the Bram Stoker Award for Best Novel in 1987, this is an epic story after a nuclear war. Similar to Stephen King's The Stand but much better written and more tightly plotted. EXCELLENT!
The Stand by Stephen King (novel) Capt. Tripps, a deadly flu ravages the Earth and the survivors deal with the aftermath. First half of the book is possibility the best thing King has done. However, it dissolves into a mish-mash of feel-good spiritualism toward the end.Trash Can Man rocks!
Doomsday Book by Connie Willis (novel) Winner the Nebula and Hugo Awards, this is a towering work of sci-fi dealing with time travel, an flu plague and medieval history.