New Netherland/New York, a colony under two different nations deserves two posts. Here's part one:
The Dutch East India Company claimed the Hudson Valley region as a result of Henry Hudson's exploration of the Atlantic Coast in 1609. The primary interest of the Dutch was the fur trade. Early on they maintained a fairly good relationship with the Five Nations [tribes] of the region. Several trading posts were established along the North, South and Fresh Rivers.
A permanent settlement was established at Fort Orange [Albany] in 1624. Many of the early settlers were Walloons, Huguenots and Africans [who were eventually able to gain 'half-free' status.]
Under new director Peter Minuit  the colony began to grow. Manhattan Island was purchased and Fort Amsterdam and the walled settlement of New Amsterdam were built. Rennselaerswyck, Beverwyck and Widwyck were also established as settlements.
Dutch settlements were set up on a patron system. The patron was granted a large tract of land and could set up a government. He was required to settle 50 families on the tract.
Peter Stuyvesant became governor/director-general in 1647. During his reign, the colony grew, He took over New Sweden and faced war with the Esopus Indians. The Peach War of 1655, so named because a settler shot an Indian for stealing peaches, led to the Cornelis May settlement on Staten Island being wiped out and several other towns being burned.
Walloon, Huguenot and Dutch settlers continued to establish towns along the North River. Staten Island was resettled in 1661.
In 1664, the English gained control of New Netherland and renamed it New York. The Dutch briefly regained control on 1673.
Family connections: Prall, Billiou, DuBois, Christopher.Christoffels, Garrisson/Gerritsen, Blom, Swart, Titsoort and a few other.