The European conquest of the New World was facilitated by a growth of exploration. Nations from Europe started to develop greater economic prosperity and started to channel these particular funds into exploration and building up an international empire. The Spanish were players in this game, followed by the English and the French. Both of these nations made significant inroads in the colonization of the new world, specifically North and South America. Both had competing interests in North America which were resolved once and for all in the French and Indian War, where Britain prevailed and ended the French hopes of prolonged presence in North America.
Europeans conquered the New World from many directions, representing mostly the British, Spanish and French empires at the time. Columbus explored and introduced the idea that Europeans were permanently interested in colonizing and conquering the land, but the first real conquest took place under the conquistadores, such as Hernan Cortes in 1519, when he began his year and a half long conquest of the Aztecs. Francisco Pizarro followed in the early 1530's with the Inca civilization in present day Peru.
Later in 1607, Jamestown was founded as the first permanent English settlement, and the French established "New France" in Canada and Louisiana in the 1650s and 1660's.
Full conquest took place under the Americans in the late 1800's as the last of Native resistance was crushed.
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Basically speaking, this conquest started in 1492 when Columbus's first voyage made the New World widely known to Europeans. This started the move towards colonizing the New World. The first major move onto the continent came in 1520 when the Spanish took what is now Mexico City from the Aztecs.
Over time, more and more of South America was taken by the Spanish and Portuguese.
Systematic colonization of North America started in 1619 (there was some colonization of Canada by the French before that, but it was more of trading posts than attempts at towns) when the Jamestown settlement was established.
From 1492 to 1518 only modest attempts were made at settlement. Three decades after 1518 became decades of conquest.
1) Nearly exterminated native peoples of the Caribbean Islands
2) Toppled and plundered the great Aztec and Inca empires
3) Gained control of territories 10 times as large as homeland
4) Discovered fabulous silver mines
5) Built an oceanic trade
1) Concentrated mostly on building an eastward oceanic trade to southeastern Asia
2) 1493: the Pope had drawn a north-south line 300 miles west of the Azores, dividing the Spanish and Portuguese spheres
3) 1494: in the Treaty of Tordesillas, Portugal obtained Spanish approval to move the line 270 leagues farther west (no one knew at the time this would give Portugal claim to Brazil)
Columbus’s second voyage to the New World in 1493 established the first Spanish colony in the Americas (in present-day Santo Domingo). The 3 million Taínos on the island, like thousands of native peoples who followed, encountered subjugation, biological disease, and ecological changes to the environment after their encounter with Spain. The Spanish left means of propagating European crops, livestock and weeds. The Spanish used military force to subdue Indians and turn them into captive labor. A similar fate befell people of Puerto Rico in 1508 and Cuba in 1511. Spanish diseases touched off a biological holocaust that killed most of the Native Americans on these islands within a single generation. Over the course of the sixteenth century, over a quarter of a million Spaniards, mainly young men, came to the Americas. Within a single generation after the death of Columbus, Spain had conquered most of South America (except Brazil), Central America and the southern parts of North America from present-day Florida to California.