How did Mexican lands in the West become part of the United States?

While conflict between the US and Mexico could be traced back to the early 19th century, perhaps initially exacerbated by western growth as a result of the Louisiana Purchase, the annexation of Mexican land really took off in the 1840s. Driven by the principles of Manifest Destiny and the inevitability of American western expansion, the US government waged a war against Mexico under President James Polk. While there was limited interest in annexation in the 1830s, and the Republic of Texas spoke to this fact, Polk was elected in 1844 on a pro-annexation platform.
The initial annexation of the Republic of Texas sparked the Mexican-American War, which lasted from 1846–1848. Early skirmishes broke out into full-blown war, although there was political debate about the validity of annexation in the US throughout the conflict. Nonetheless, the US Army was already quite well-equipped and organized by this point and pushed Mexican troops all the way back to the capital Mexico City. There was also another significant military campaign fought in what was the Republic of California.
Outgunned and facing American occupation, the Mexican government had no choice but to sue for peace. This resulted in signing the Treaty of Guadalupe-Hidalgo in early 1848. This treaty ceded former Mexican territory to the US. This so-called Mexican Cession was to become most of the southwestern US as settlement continued in the late 19th century. A small amount of further territory was acquired in the Gadsden Purchase of 1853.