In the 18th century and early 19th century many writers and antiquitarians believed that various Old World cultures were responsible for the ancient monuments found in the New World. Part of this was due to ethnocentrism, for many European and European-American writers did not believe that Native Americans were capable the degree of civilization required to build the ancient monuments of the Americas.
In the 1830s and 1840s better information became available and changed popular opinion. (Most notable were the detailed first hand descriptions of ancient monuments of Mesoamerica by John Lloyd Stephens and accounts of the Spanish conquests of Mexico and Peru compiled by William H. Prescott based on Spanish colonial documents. The work of both writers sold well.) After evaluating such new data, most historians came to the conclusion that the ancient monuments of the Americas were built by the ancestors of the current Native Americans.
For about a century, the possibility of Pre-Columbian trans-Atlantic contacts was usually dismissed by most mainstream historians. So far as Westerners were concerned, the Americas had been "discovered" by Christopher Columbus in 1492, and no previous voyages were likely: trans-Atlantic and trans-Pacific voyages were thought unlikely before the age of European exploration starting in the late 15th century, and most historians considered that there was no reliable evidence that any such contacts had taken place.
In recent decades it has become clear that there have been many long ocean voyages prior to the European explorations. Linguistic evidence demonstrates that Madagascar was settled by Austronesian peoples from Indonesia, for example: their navigators were able to cross the Indian Ocean and large sections of the Pacific more than a millennium ago.
In the mid-20th century, several attempts were made to demonstrate the possibility of long survival on ocean voyages, like that of Alain Bombard. Norwegian writer Thor Heyerdahl used light reed boats (named Rá and Kon-Tiki), similar to those used in North Africa and Bolivia. By successfully crossing both the Atlantic and Pacific Oceans with these craft, Heyerdahl demonstrated that there was no technical reason why the Americas could not have been settled from Africa, or the Pacific Islands from South America. The archeological significance if any of such voyages is debated, but these demonstrations made many of the general public curious about Pre-Columbian trans-oceanic contact.
Some ancient Viking chronicles talked about a land called Vinland to the west of Greenland. Historians debated the meaning of these chronicles, and whether the Vikings had ever visited the New World in Pre-Columbian times. These debates were settled by archeological evidence in 1961. In that year Helge Ingstad and Anne Stine Ingstad discovered the remains of a Viking settlement at L'Anse aux Meadows in Newfoundland. This was clear proof that the Vikings had crossed the Atlantic by way of Iceland and Greenland around 1000 CE.
Various artifacts which some think suggest other Pre-Columbian trans-Atlantic contact has been found, but none presenting such clear evidence as L'Anse aux Meadows. Most archeologists and historians continue to hold that contacts between other Pre-Columbian Old and New World civilizations should be considered unlikely unless better evidence to the contrary surfaces.
As was the case in the early 19th century, there are writers today who claim that Old World civilizations such as those of Israel, Egypt, Irish monks (as hinted by the legend of St Brendan), Ancient Rome, Islamic West Africa, Sumeria, the Temple Knights, etc had landed on the Pre-Columbian Americas. (Trans-Pacific influences have also been proclaimed, as have contacts with Atlantis and other supposed "lost continents"; in the 20th century extra-terrestrial civilizations have been added to the long list of the suggested "real" builders of the ancient monuments of the Americas.) Such writings are considered at least very dubious if not simply contrary to known facts by mainstream historians and archeologists.
There is also the hypothesis of a prediscoverer in the 15th century whose testimony had inspired Columbus. The secret proofs of Columbus would explain how he convinced the Catholic Kings in spite of being actually wrong about the position of Asia, as the royal experts knew.
Another hypothesis tells that Basque sailors, whose presence in North America is documented from just years after Columbus, could have reached the land earlier while on their cod and whaling campaigns.
See also:
Some books dealing with pre-Columbian crossings: