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After expanding aggressively and conquering even more aggressively, you should have a good sized empire. That does not make you invincible, however. Opponents may gang up and you to try and take you down, or another civilization on the other side of the world may be conquering cities as quickly as you are.
That means it is time to start researching solutions to your long term problems. The two major long-term problems are transportation and military might. It is important to always be expanding, but lack of transportation technology often brings an offensive to a stand-still. It is difficult to move armies if there are no roads to use, and even more difficult to move an army over water. By the time you've conquered one civilization, water is likely becoming an obstacle in at least one direction, so be sure to vigorously research naval technologies.
Also be sure to research the latest and greatest military technologies. There are certain technologies, like gunpowder units and aircraft, which completely change the way the game is played. Reaching these technologies first is crucial. If you maintain the technological edge when it comes to military units, victory is almost guaranteed.