I would argue, while not disagreeing with any of the above posts, that decolonization was among the most important events of the post World War II period. Millions, even billions, of people around the world became part of independent nations instead of colonial subjects, though this process did not proceed smoothly. Indeed, the majority of the world's poorest and most politically unstable nations are former colonies, and they struggle to overcome many of the legacies of colonialism. Still, the emergence of nations such as India, Pakistan, and others was an astonishing event.

The fall of communism around the world, as exemplified by the fall of the Berlin Wall and the unification of Germany would certainly strike me as an important world event. As did a previous responder, I also grew up in the time of bomb shelters, "duck and cover" drills, fear of when (not if) a nuclear war would break out. All of this, of course, was punctuated by the Cuban Missile Crisis. We truly lived in a bi-polar world at that time, and no solution short of war was on the horizon. The fact that these trying times ended, in T.S. Eliot's words, "not with a bang, but a whimper," is truly significant.

There are two events that I always remember.  One of them is the Challenger space shuttle explosion.  I remember watching it on TV thinking how exciting it was that they were sending a teacher into space when the whole thing exploded.  The second event is September 11th.  I had several friends in the military at that point and I was frightened they would be sent to war.  It scary to wonder when the attacks would stop or if there would be more towns targeted.  In my own town, we have a large portion of the oil pipeline and other significant resources.  I remember them placing tanks and armed guards in the streets around the oil holding tanks.  A lot of people don't realize how the military responded to protect the country and its resources in places apart from New York and Washington.

Since the question asks for "World Events," the collapse of was epic for those of us who remember the Fallout Shelter drills and the Cuban Crisis. The "end of Communism" is a historical event that was momentous at the time, but now with former KGB Putin as Prime Minister, one wonders.... 

I wholeheartedly agree with all the posts above, and I would add Sept. 11, 2001 to the list. I think the shock of that event and the repercussions that followed make it the most memorable event for me. I believe it will forever alter how we view ourselves and our country from a global perspective. It led to two wars of debatable value, and that will affect our future also.

As a person who is multi-racial, I would say that one of the biggest events since 1945 was the Civil Rights Movement.  When my parents got married in 1964, their marriage would still have been illegal in many states in the South.  They probably wouldn't have been able to eat together at many restaurants in the region.  The changes brought about by the movement have made my life and that of my children much easier than it would have been in those times.

The fall of the Berlin wall and the collapse of the USSR are two significant happenings from the late 20th century. Communism, as it once was, and the Cold War effectively ended with the symbolic collapse of the wall separating East Berlin from West Berlin, pushing back the Iron Curtain from central Europe (until it eventually ceased to exist). 

For my age, I will always remember the falling of the Berlin Wall and the end of the Cold War.  My formative years involved viewing everything from the news to the Olympics as a struggle of democracy vs. communism.  That change in the geopolitical paradigm has always been one of the most important to me personally.

I think one of the most significant events, and one I distinctly remember, was the landing on the moon by Apollo 11 on July 20, 1969. I remember the news clips and live broadcasts on the television set for days. I watched the pre-launch events, the actual launch, the joining of the lunar module with the command module, the landing and take-off, and the reentry. It was almost surreal to watch, and no one that I know of who saw it happen on television will ever forget it as long as they live!