Territorial expansion in the U.S. began following American Independence and continued rapidly through to the 1860s. Following this period, the U.S. did acquire other territories, but at a slower rate.
Following the 1783 Treaty of Paris, the U.S. had strictly defined borders. However, Thomas Jefferson negotiated for the State of Louisiana in 1803, and other southern states soon followed. A rapid period of expansion ensued in the 1840s, including the acquisition of Maine and Texas.
After its successes in the 1840s, the U.S. attempted to further its expansion in Central America and Cuba. These campaigns were mainly unsuccessful, although areas like Alaska and Hawaii were acquired later in the century.