London
New York
Liverpool
Philadelphia
Dublin
Edinburgh
Paris (maybe second tier) 
I don’t see how Milan gets into the wiki list either.
Jerusalem perhapse should be in for it’s historical and religious importance.
Other possibles Budhapest, Barcellona, St. Petersberg,.
Okay, after some thought, here is how I tackled this question - if I wanted to show a friendly alien the most important examples of urban Earth, I would take it to the following cities, in no particular order:
New York City
Chicago (okay, I’m biased!)
San Francisco (okay, biased again, since it’s my hometown)
London
Jerusalem
Mecca
Cairo
Shanghai
Mexico City
Tokyo
Sydney
Rome
Bombay
Johannesburg
Rio de Janeiro
Moscow
Bah, not even the greatest city in it’s state.
And Green Bay needs to be added because of the Packers.
Um…no. Green freaking Bay?
What, you can’t possibly be thinking that San Diego is better than San Francisco. What do they have there? Okay, the zoo is nice.
London
Paris
Bristol
Munich
In that order.
I have to disagree on Tokyo. It’s just too damn big and crowded and difficult to get around.
If there is one truly outstanding city in Japan, it’s Kyoto.
It’s absolutely gorgeous, has wonderful history and architecture, a first rate university, and a visit won’t make you bankrupt.
What about the entire non-Europe part of the world?
Not San Diego, I’m talking about Bakersfield!
Oh well, yeah, Bakersfield is pretty awesome. What with the…dirt, and all.
Have the people who have said Rio been to a different Rio than I have? The Rio I was at was overcrowded, filthy and crime ridden. I don’t deny that some parts of the city are fantastic, and they do have fabulous beaches with lots of hotties in tiny bikinis, but that doesn’t qualify it as a great city of the world.
I’m going to limit this to 10 cities I’ve lived in or visited, and there’s nothing objective about this list. In order:
Paris (tie)
NYC (tie)
Rome
Vienna
London
San Francisco
Venice
Montreal
Boston
Florence
Bristol ??? Well that is an interesting choice, and historically it is a very important port, but what do you say makes Bristol world class as a city?
I have lived in Bristil and worked in Brislington, but never saw any reason to think Bristol was a world class city.
Well aren’t parts ofNew York and London “overcrowded, filthy and crime ridden” (and just about every other choice listed above)? Is Rio really worse than NYC or London in those areas?
Don’t know much about Rio, but it should be noted the OP never mentioned anything about how liveable or orderly or clean the cities are. Rather, the criteria were power, beauty, culture, and history.
From places I’ve been to:
Rome
Paris
London
NYC
Venice
The cities I have been to, and would go back for more:
San Francisco
Seattle
Austin
New Orleans
Miami
NYC
Toronto
Amsterdam
Berlin
Dublin
London
Stockholm
How about Los Angeles as one of the greatest cities in the world? It may not be the #1 tourist destination, but as a meeting place between Asia, Central/South America and the United States, it is one of the most important cities in the world.
I love Los Angeles as a city, but it isn’t where I’ll always take my vacation. However as a great world city, it may only topped by a few. Perhaps New York, London, Paris, Rome…few others, I think.
Based on the criteria of power, beauty, culture, history as stated in the OP, and restricted to only cities in which I’ve lived or have visited:
[ul]
[li]Paris[/li][li]London[/li][li]Rome[/li][li]NYC (on the East Coast)[/li][li]San Francisco (on the West Coast)[/li][li]Hong Kong[/li][/ul]
Other cities I’ve seen, but do not include on my “greatest” list:
[ul]
[li]Los Angeles … where’s the beauty? Power? Both U.S. Senators are from the SF area, and the state is deep blue due to the northern part of the state[/li][li]Barcelona … entertaining city, but a bit too grimy for me[/li][li]Venice … great tourist city, that’s it[/li][li]Florence … love the city, but I’m only picking one city from Italy[/li][li]Amsterdam … very nice tolerant city, wonderful for walking, just shy of being one of the greatest[/li][li]Vancouver and Victoria, BC … gorgeous cities, but can’t put these alongside the ones above[/li][/ul]
Other cities I will visit and might make my “greatest” list post-visit:
[ul]
[li]Beijing[/li][li]Shanghai[/li][li]Istanbul[/li][li]Mexico City[/li][li]Sydney[/li][/ul]
The Top Ten Cities In The World
to me…
Madison, WI
Asheville, NC
Columbia, MD
Kitty Hawk, NC
Savannah, GA
Vicenza, Italy
Leusden, The Netherlands
King’s Mountain, NC
Key West, FL
Baltimore, MD
Sadly, I have not traveled the world enough to know.
But, here’s the top 10 I’ve visited and loved or would like to visit, though I’m not sure they’re all “cities”:
Christchurch, New Zealand
Palermo, Sicily
Phuket, Thailand
Trondheim, Norway
New York City, NY
Vancouver, BC
Montreal, QB
Dusseldorf, Germany
St. Petersburg, Russia
Washington DC