Using two comprehensive datasets on population of cities (1800-2000) and metropolitan areas (1960-2000) for a large set of countries, I present three new empirical facts about the evolution of city growth. First, the distribution of cities growth rates is skewed to the right in most countries and decades. Second, within a country, the average rank of each decade's fastest growing cities tends to increase over time. Finally, this rank grows faster in periods of rapid growth in urban population. These facts can be interpreted as evidence in favor of the idea that urban agglomerations have historically grown following a sequential growth pattern: within a country, the initially largest city is the first one to grow rapidly for some years. At some point, the growth rate of this city slows down and the second largest city is then the fastest-growing one. Eventually, the third largest city starts growing fast as the two largest cities slow down, and so on.